[\Cref{main}]
If $(A,\sigma)$ is a totally decomposable $K$-algebra with involution of the first kind with $\ind A\leqslant 2$, then ${\bf PSim}^+(A,\sigma)$ is $R$-trivial.

Let $K$ be a field of characteristic different from $2$.
Let $(A,\sigma)$ be a $K$-algebra with involution. Let ${\bf PSim}(A,\sigma)$ denote the algebraic group of its projective similitudes and ${\bf PSim}^+(A,\sigma)$ the connected component of the identity.
The latter is a classical semi-simple connected linear algebraic group of adjoint type.
(In various cases, one has ${\bf PSim}^+(A,\sigma)={\bf PSim}(A,\sigma)$, e.g.~if $\sigma$ is symplectic.)
In \cite{Mer96}, Merkurjev studied
 the geometric properties of ${\bf PSim}^+(A,\sigma)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.

In this article, we assume that the involution $\sigma$ is $K$-linear (i.e.~an involution of the first kind).
We call $(A,\sigma)$ \emph{totally decomposable}
if $$(A,\sigma)\simeq \Ad(\psi)\otimes\left(\bigotimes_{i=1}^r (Q_i,\tau_i)\right)$$
for some $r\in\mathbb{N}$, $K$-quaternion algebras with involution $(Q_i,\tau_i)$ for $1\leqslant i\leqslant r$ and the adjoint algebra with involution $\Ad(\psi)$ of some odd-dimensional quadratic form $\psi$ over $K$. (Taking $\psi=\langle 1\rangle$ corresponds to the situation without the first factor.)

If $\sigma$ is orthogonal and $A$ is split, then we obtain by \cite[Theorem 1]{Bec08} that $(A,\sigma)\simeq \Ad(\psi\otimes \pi)$
for some Pfister form $\pi$ over $K$, and in this case it follows by \cite[Prop.~7]{Mer96} that ${\bf PSim}^+(A,\sigma)$ is stably rational, and in particular $R$-trivial, by \cite[Prop.~1]{Mer96}.
One may ask whether these facts still hold without assuming that $A$ is split.
In this article, we show:

\begin{cor}\label{main}
Let $(A,\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\ind A\leq 2$. Then ${\bf PSim}^+(A,\s)$ is $R$-trivial.
\end{cor}

\begin{abstract}
We show that the group of proper projective similitudes of a totally decomposable algebra with involution of the first kind over a field of characteristic different from $2$ is $R$-trivial.

Let $K$ be a field of characteristic different from $2$.
Let $(A,\s)$ be a $K$-algebra with involution. Let ${\bf PSim}(A,\s)$ denote the algebraic group of its projective similitudes and ${\bf PSim}^+(A,\s)$ the connected component of the identity.
The latter is a classical semi-simple connected linear algebraic group of adjoint type.
(In various cases, one has ${\bf PSim}^+(A,\s)={\bf PSim}(A,\s)$, e.g.~if $\s$ is symplectic.)
In \cite{Mer96}, Merkurjev studied
 the geometric properties of ${\bf PSim}^+(A,\s)$ as a variety, regarding the question whether it is rational, stably rational or $R$-trivial.

In this article, we assume that the involution $\s$ is $K$-linear (i.e.~an involution of the first kind).
We call $(A,\s)$ \emph{totally decomposable}
if $$(A,\s)\simeq \Ad(\psi)\otimes\left(\bigotimes_{i=1}^r (Q_i,\tau_i)\right)$$
for some $r\in\nat$, $K$-quaternion algebras with involution $(Q_i,\tau_i)$ for $1\leq i\leq r$ and the adjoint algebra with involution $\Ad(\psi)$ of some odd-dimensional quadratic form $\psi$ over $K$. (Taking $\psi=\la 1\ra$ corresponds to the situation without the first factor.)

If $\s$ is orthogonal and $A$ is split, then we obtain by \cite[Theorem 1]{Bec08} that $(A,\s)\simeq \Ad(\psi\otimes \pi)$
for some Pfister form $\pi$ over $K$, and in this case it follows by \cite[Prop.~7]{Mer96} that ${\bf PSim}^+(A,\s)$ is stably rational, and in particular $R$-trivial, by \cite[Prop.~1]{Mer96}.
One may ask whether these facts still hold without assuming that $A$ is split.
In this article, we show:

The most interesting case is that where $\s$ is orthogonal and $\ind A=2$.
To our knowledge, this is the first general positive result on $R$-triviality of ${\bf PSim}^+(A,\s)$ that applies to non-split algebras with orthogonal involution $(A,\s)$ of degree divisible by $8$.

Let $(A, \sigma)$ be an $K$-algebra with involution of the first kind. (That is, $A$ is a central simple $K$-algebra and $\s$ is a $K$-linear involution on $A$.)
In $\mg{K}$, we consider the subgroup
$$\G(A,\s)=\{\s(x)x\mid x\in\mg{A}\}\cap\mg{K}\,.$$
Its elements are called \emph{similarity factors} or \emph{multipliers of similitude of $(A,\s)$}. We refer to \cite[\S12]{KMRT98} for a discussion of similitudes and their multipliers.

\begin{thm}\label{P:main}
 Let $(A,\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind of even degree and with $\ind A\leq 2$.
 For every $c\in\G(A,\s)$ there exist $d_1,d_2\in\mg{K}$ such that $\s_{K(\sqrt{d_i})}$ is hyperbolic for $i\in\{1,2\}$ and $c\in\D\lla d_1\rra\cdot \D\lla d_2\rra$.
\end{thm}
\begin{proof}
It follows from the hypothesis by \cite[Corollary]{Bec08} if $\s$ is symplectic and by \cite[Theorem 2]{Bec08} if $\s$ is orthogonal that there exists a decomposition
$$(A,\s)\simeq \Ad(\psi\otimes \rho)\otimes (Q,\gamma)$$
with a $K$-quaternion algebra with involution of the first kind $(Q,\tau)$, a Pfister form $\rho$ over $K$ and an odd-dimensional quadratic form $\psi$ over $K$.

\begin{cor}\label{main}
Let $(A,\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\ind A\leq 2$. Then ${\bf PSim}^+(A,\s)$ is $R$-trivial.
\end{cor}
\begin{proof}
Using \cite[Theorem 1]{Mer96}, this follows immediately from \Cref{P:main}.
\end{proof}

\begin{cor}\label{main}
Let $(A,\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\ind A\leq 2$. Then ${\bf PSim}^+(A,\s)$ is $R$-trivial.
\end{cor}

The most interesting case is that where $\sigma$ is orthogonal and $\ind A=2$.
To our knowledge, this is the first general positive result on $R$-triviality of ${\bf PSim}^+(A,\sigma)$ that applies to non-split algebras with orthogonal involution $(A,\sigma)$ of degree divisible by $8$.

For the proof, we crucially rely on the characterization of $R$-equivalence in ${\bf PSim}^+(A,\sigma)$ established in \cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\times}$ generated by the norms from finite field extensions where the involution $\sigma$ becomes hyperbolic.
The main part of our proof consists in describing the similarity factors of $(A,\sigma)$ in terms of quadratic forms over $K$.
To this end, we combine a variety of key results from quadratic form theory.

For the proof, we crucially rely on the characterization of $R$-equivalence in ${\bf PSim}^+(A,\sigma)$ established in \cite[Theorem 1]{Mer96}. It translates the question into that of comparing the group of similarity factors with the subgroup of the multiplicative group ${K}^{\times}$ generated by the norms from finite field extensions where the involution $\sigma$ becomes hyperbolic. The main part of our proof consists in describing the similarity factors of $(A,\sigma)$ in terms of quadratic forms over $K$. To this end, we combine a variety of key results from quadratic form theory.,

\begin{cor}\label{main}
Let $(A,\sigma)$ be a totally decomposable $K$-algebra with involution of the first kind such that $\ind A\leq 2$. Then ${\bf PSim}^+(A,\s)$ is $R$-trivial.
\end{cor}
If $(A,\sigma)$ is a totally decomposable $K$-algebra with involution of the first kind with $\ind A\leqslant 2$, then ${\bf PSim}^+(A,\sigma)$ is $R$-trivial.,

Question:
Let $K$ be a field of characteristic different from $2$. A $K$-algebra with involution of the first kind means a central simple $K$-algebra $A$ equipped with a $K$-linear involution $\sigma$. Call $(A,\sigma)$ totally decomposable if
$$
(A,\sigma)\simeq \operatorname{Ad}(\psi)\otimes\left(\bigotimes_{i=1}^r (Q_i,\tau_i)\right)
$$
for some $r\in\mathbb N$, some $K$-quaternion algebras with involution $(Q_i,\tau_i)$, and the adjoint algebra with involution $\operatorname{Ad}(\psi)$ of an odd-dimensional quadratic form $\psi$ over $K$. Let ${\bf PSim}(A,\sigma)$ denote the algebraic group of projective similitudes of $(A,\sigma)$, and let ${\bf PSim}^+(A,\sigma)$ be its connected component of the identity. Which statement holds for every totally decomposable $K$-algebra with involution of the first kind satisfying $\operatorname{ind} A\le 2$?

Choices:
B. ${\bf PSim}(A,\sigma)$ is $R$-trivial.
C. ${\bf PSim}^+(A,\sigma)$ is unirational over $K$.
D. For every such $(A,\sigma)$, the variety ${\bf PSim}^+(A,\sigma)$ is stably rational.
E. For every such $(A,\sigma)$ and every $c\in K^{\times}$, there exist quadratic extensions $L_1/K$ and $L_2/K$ over which $\sigma$ becomes hyperbolic such that $c\in N_{L_1/K}(L_1^{\times})\,N_{L_2/K}(L_2^{\times})$; hence ${\bf PSim}^+(A,\sigma)$ is $R$-trivial.

The stem does not explicitly reveal the correct option. It states hypotheses and definitions, but it does not directly announce that ${\bf PSim}^+(A,\sigma)$ is $R$-trivial.

This is essentially a direct theorem-recall item: the hypotheses are stated in full, and the correct choice is just the theorem's conclusion restated almost verbatim.

There is some pressure to distinguish the exact conclusion from nearby variants (split-only, finite quotient, after extension, full group vs identity component), but the item mainly tests recognition/recall rather than genuine derivation or synthesis.

The distractors are mathematically plausible and target realistic confusion points: weakening the conclusion, adding unnecessary split hypotheses, shifting to a field extension, or confusing ${\bf PSim}^+$ with the full ${\bf PSim}$.

A mathematically well-constructed recall question with strong distractors, but it is largely theorem restatement rather than a generative reasoning task.


Question:
Let $K$ be a field of characteristic different from $2$. A $K$-algebra with involution of the first kind means a central simple $K$-algebra $A$ equipped with a $K$-linear involution $\sigma$. Call $(A,\sigma)$ totally decomposable if
$$
(A,\sigma)\simeq \operatorname{Ad}(\psi)\otimes\left(\bigotimes_{i=1}^r (Q_i,\tau_i)\right)
$$
for some $r\in\mathbb N$, some $K$-quaternion algebras with involution $(Q_i,\tau_i)$, and the adjoint algebra with involution $\operatorname{Ad}(\psi)$ of an odd-dimensional quadratic form $\psi$ over $K$. Let ${\bf PSim}(A,\sigma)$ denote the algebraic group of projective similitudes of $(A,\sigma)$, and let ${\bf PSim}^+(A,\sigma)$ be its connected component of the identity. Which statement holds for every totally decomposable $K$-algebra with involution of the first kind satisfying $\operatorname{ind} A\le 2$?

Choices:
B. ${\bf PSim}(A,\sigma)$ is $R$-trivial.
C. ${\bf PSim}^+(A,\sigma)$ is unirational over $K$.
D. For every such $(A,\sigma)$, the variety ${\bf PSim}^+(A,\sigma)$ is stably rational.
E. For every such $(A,\sigma)$ and every $c\in K^{\times}$, there exist quadratic extensions $L_1/K$ and $L_2/K$ over which $\sigma$ becomes hyperbolic such that $c\in N_{L_1/K}(L_1^{\times})\,N_{L_2/K}(L_2^{\times})$; hence ${\bf PSim}^+(A,\sigma)$ is $R$-trivial.

The stem does not explicitly reveal the correct option, nor does it contain a near-verbatim hint of the conclusion. The answer must be identified from the options rather than extracted from the wording.

This is essentially a theorem-recall item: the hypotheses are stated in full and the student is asked which conclusion holds for all such objects. That makes it very close to a direct restatement of a known result.

There is some reasoning pressure because the options are nearby variants: connected component vs full group, R-triviality vs unirationality vs stable rationality, and a technical norm-condition overreach. Still, the item mainly tests recognition/recall rather than genuine derivation.

The distractors are mathematically plausible and target realistic failure modes: confusing the connected component with the full group, replacing the correct statement by a weaker true one, strengthening to stable rationality, or overgeneralizing proof ingredients into an all-purpose norm statement.

A mathematically well-constructed theorem-discrimination MCQ with strong distractors and little answer leakage, but it is largely a direct recall/restatement question rather than one that strongly tests generative reasoning.


Assume that (F1-5) hold true.
    Then there exists a nontrivial periodic solution for~\eqref{eq:NDE-spatial}, which is in~$C^1$.

\begin{rmk}\label{rmk:multiplicity}
    It is tempting to hope for multiple solutions at this stage, especially when~$F$ is even in~$\psi$ or has more symmetries, as in~\cite{DingRuf2008Solutions} and~\cite{DingLiu2014periodic}.
    This would require a restrcition of the value of~$a$, or equivalently, that~$m\gg a$.
    The latter condition on~$m$ is not too restrictive since, in the orginal setting, we should have~$mc^2$ where~$c$ stands for the speed of light.
    There are plenty of works on the limiting behavior as~$c\to+\infty$.
    We leave this, together with the multiplicity, to a future work.
\end{rmk}

In physical models for fermionic fields, the equations of motion often appear as Dirac equations with various nonlinearities.
In this work we consider periodic solutions on~$\mathbb{R}^{1+3}$, which are actually spatially periodic solutions and stationary in physical sense i.e. they propagate via a complex phase factor without changing shapes.
Such solutions fails to decay at infinity, and we have to employ a perturbative variational scheme to obtain a nontrivial solution.

\

More precisely, we will be concerned with the following nonlinear Dirac equation
\begin{align}\label{eq:4D nonlinear Dirac}
    \sum_{\mu=0}^{3} i\gamma^\mu\p_\mu\Psi-m \Psi + \gamma^0 \nabla F(\Psi)=0
\end{align}
where~$\Psi\colon \mathbb{R}^3\to \mathbb{C}^4$ stands for a spinor field,~$m>0$ corresponds to the mass of the particles, and~$F\colon \mathbb{C}^4\to \mathbb{R}$ is the self-coupling of particles in the system, which are chosen according to the complexity of the physical and chemical models, see e.g.~\cite{BartschDing2006Solutions,Ding2007variational, Esteban2002overview,Psarelli2005Maxwell,Soler1970Classical, Thaller1992Dirac} and the references therein.
The~$\gamma^\mu$'s are as given follows.
The matrix~$\gamma^0$ takes the form
\begin{align}
    \gamma^0= \begin{pmatrix} I_2 & 0 \\ 0 & -I_2\end{pmatrix}
    =\begin{pmatrix}
        1 & & &  \\ & 1 & & \\ & & -1 & \\ & &  & -1
    \end{pmatrix}
\end{align}
and induces a decomposition of the spinors into~$\pm1$ eigenspaces of~$\gamma^0$.
As for the others, we denote by
\begin{align}
    {\sigma^1} = \left( {\begin{array}{*{20}{c}}
			{0}&1 \\
			1&{0}
	\end{array}} \right),\;{\sigma^2} = \left( {\begin{array}{*{20}{c}}
			{0}&{ - i} \\
			i&{0}
	\end{array}} \right),\;{\sigma^3} = \left( {\begin{array}{*{20}{c}}
			1&{0} \\
			{0}&{ - 1}
	\end{array}} \right)
\end{align}
for the Pauli matrices, and the gamma matrices are
\begin{align}
    {\gamma^k} = \left( {\begin{array}{*{20}{c}}
			{0}&{{\sigma _k}} \\
			{{-\sigma _k}}&{0}
	\end{array}} \right),\quad\text{for}\quad k= 1, 2, 3.
\end{align}
Note that they satisfy the following Clifford relations: for~$j,k\in\left\{1,2,3\right\}$,
\begin{align}
    \sigma^j\sigma^k+\sigma^k\sigma^j= 2\delta^{jk}I_2, & &
    \gamma^j\gamma^k + \gamma^k\gamma^j= -2\delta^{jk}I_4.
\end{align}
Moreover,
\begin{align}
    \gamma^0\gamma^k + \gamma^k \gamma^0 =0, & &
    (\gamma^0)^2=I_4.
\end{align}
Then~$\sum_{\mu} i\gamma^\mu \p_\mu$ is the 4D space-time Dirac operator.
With the time direction involved, it is not elliptic.

An typical ansatz for the solutions are the so-called \emph{stationary solutions} or~\emph{solitary solutions} (see e.g.~\cite{EstebanSere1995stationary}) which takes the form
\begin{align}
    \Psi(t,\vec{x})= e^{-i a t}\psi(\vec{x})
\end{align}
Such solutions are regarded as the “particle-like solutions” and hence attract much attention.
For spinors of such form and for~$F$ invariant under rotations ( i.e. ~$F(e^{i\theta}\Psi)=F(\Psi)$),~\eqref{eq:4D nonlinear Dirac} is transformed into
\begin{align}
    i\gamma^k\p_k\psi + a\gamma^0\psi-m\psi +\gamma^0\nabla F(\psi)=0.
\end{align}
Equivalently, we are led to consider the equation
\begin{align}\label{eq:NDE-spatial}\tag{*}
    -i\gamma^0 \gamma^k\p_k \psi + m\gamma^0\psi -a\psi-\nabla F(\psi)=0.
\end{align}
Equations of this form have been studied in depth in literature, see e.g.~\cite{BalabaneCazenave1988Existence, Balabane1988Existence, Balabane1990Existence, Cazenave1986Existence,EstebanSere1995stationary,Merle1988Existence} and the references quoted therein.
Different from~\cite{EstebanSere1995stationary}, here we are concerned with periodic solutions, namely solutions satisfying
\begin{align}
    \psi(\theta+\tau)=\psi(\theta), \quad \forall \tau\in \Gamma
\end{align}
where~$\Gamma\subset \mathbb{R}^3$ is a 3-dimensional lattice.
They are referred as the standing periodic waves for~\eqref{eq:4D nonlinear Dirac}.

\

In quantum fields theory it is often required that the Hamiltonian has some symmetry, and the periodicity is thus frequently imposed.
In one-dimensional dynamics this is usually the case.
According to the features of the particle system, sometimes it is require that the nonlinearity has spatial periodicity, which is not the case here since our~$F$ has no explicit dependence on the space variable~$x$.
Here we are in a similar setting as in~\cite{DingLiu2014periodic}, namely we seek for periodic solutions for the nonlinear Dirac equation~\eqref{eq:NDE-spatial}.
The difference to ~\cite{DingLiu2014periodic} lies in the nonlinearity, which will become clear later.

Geometrically, this amounts to consider the equation~\eqref{eq:NDE-spatial} on the compact manifold~$\mathbb{T}^3=\faktor{\mathbb{R}^3}{\Gamma}$.
The spectral properties for the linearized operators on torus is in great contrast to the one on~$\mathbb{R}^3$, so is the analysis needed for the variational scheme.
For example, we cannot appeal to the Fourier space to obtain enough linking structures, and we do not have a global Pohozaev identity to control the kinetic energy and the potential energy separately, which are crucial for the proof in~\cite{EstebanSere1995stationary} to obtain solutions on~$\mathbb{R}^3$.
On the other hand, the torus is still good enough to allow for variational analysis and we can changle local strategies to carry out the variational method in~\cite{EstebanSere1995stationary} to obtain nontrivial solutions.

\

The main motivation of the nonlinearities comes from the following examples.
In the Soler model~\cite{Soler1970Classical}, the self-coupling has the form
\begin{align}\label{eq:F Soler}
    F(\psi)
    =\frac{1}{2}G(\bar{\psi}\psi)
\end{align}
with~$G\in C^2(\mathbb{R},\R_+)$, and~$G(s)=0$ for~$s\leq 0$.
To solve the corresponding nonlinear Dirac equation on~$\mathbb{R}^3$, a suitable “radial” Ansatz were used to reduce the equation to ODEs and then the shooting method can provide some solutions.
These methods cannot be applied in the periodic setting for the lack of a good radail Ansatz.
A more general form of~$F$ arise in~\cite{Finkelstein1951Non-linea} where the physicists were studying the symmetric coupling between nucleons, muons, and leptons, for which
\begin{align}\label{eq:nonlinearity Finkelstein}
    F(\psi) = \frac{1}{2}|\bar{\psi}\psi|^2 + b|\bar{\psi}\gamma^5\psi|^2
\end{align}
where~$\gamma^5= \gamma^0\gamma^1\gamma^2\gamma^3$.
In such models, the existence theory remains still largely open.
Indeed, for~\eqref{eq:nonlinearity Finkelstein}, it has growth order~$|\psi|^4$ which is super-critical for the Sobolev embeddings, thus not appropriate for the variational scheme.

\

Up to now we can only deal with the sub-critical case.
To be precise, we impose the following hypothese on the nonlinearity~$F$.
Suppose that there exist constants~$A_j>0$,~$1\leq j\leq 4$, and~$\nu>1$,~$\alpha>2$,~$\beta>3$,~$2<\alpha_1\leq\alpha_2<3$ such that
\begin{itemize}
    \item[(F1)] \( 0 \leq F(\psi) \leq A_1 \big( |\psi|^{\alpha_1} + |\psi|^{\alpha_2} \big) \quad \forall \psi \in \mathbb{C}^4 \),

    \item[(F2)] \( F \in C^2 \), \( F(0) = F'(0) = F''(0) = 0 \), and
          \( |F''(\psi)| \leq A_2 |\psi|^{\alpha_2 - 2} \) for \( |\psi| \) large,

    \item[(F3)] \( \alpha F(\psi) \leq dF(\psi)[\psi] \) for some \( \alpha > 2 \),

    \item[(F4)] \( F(\psi) \geq A_3 |\psi\overline{\psi}|^\nu - A_4 \),

    \item[(F5)]  $\exists\quad\beta > 3$, $\forall\quad\delta > 0$, $\exists \quad C_\delta > 0$, $\forall \psi \in \mathbb{C}^4$, such that
\begin{align}
     |\nabla F(\psi)| \leq \left( \delta + C_\delta F(\psi)^{\frac{1}{\beta}} \right) |\psi|.
\end{align}
\end{itemize}
The practical nonlinearity that we aim to model include the Soler nonlinearity in~\eqref{eq:F Soler} as well as
\begin{align}
    F(\psi)= |\bar{\psi}\psi|^\alpha+ b|\bar{\psi}\gamma^5\psi|^\beta,
\end{align}
where~$\gamma^5=\gamma^0\gamma^1\gamma^2\gamma^3$ and~$b\geq 0$,~$1<\alpha,\beta<\frac{3}{2}$.

The main result of this work is the following.

An typical ansatz for the solutions are the so-called \emph{stationary solutions} or~\emph{solitary solutions} (see e.g.~\cite{EstebanSere1995stationary}) which takes the form
\begin{align}
 \Psi(t,\vec{x})= e^{-i a t}\psi(\vec{x})
\end{align}
Such solutions are regarded as the “particle-like solutions” and hence attract much attention.
For spinors of such form and for~$F$ invariant under rotations ( i.e. ~$F(e^{i\theta}\Psi)=F(\Psi)$),~\eqref{eq:4D nonlinear Dirac} is transformed into
\begin{align}
 i\gamma^k\p_k\psi + a\gamma^0\psi-m\psi +\gamma^0\nabla F(\psi)=0.
\end{align}
Equivalently, we are led to consider the equation
\begin{align}\label{eq:NDE-spatial}\tag{*}
 -i\gamma^0 \gamma^k\p_k \psi + m\gamma^0\psi -a\psi-\nabla F(\psi)=0.
\end{align}
Equations of this form have been studied in depth in literature, see e.g.~\cite{BalabaneCazenave1988Existence, Balabane1988Existence, Balabane1990Existence, Cazenave1986Existence,EstebanSere1995stationary,Merle1988Existence} and the references quoted therein.
Different from~\cite{EstebanSere1995stationary}, here we are concerned with periodic solutions, namely solutions satisfying
\begin{align}
 \psi(\theta+\tau)=\psi(\theta), \quad \forall \tau\in \Gamma
\end{align}
where~$\Gamma\subset \mathbb{R}^3$ is a 3-dimensional lattice.
They are referred as the standing periodic waves for~\eqref{eq:4D nonlinear Dirac}.

In quantum fields theory it is often required that the Hamiltonian has some symmetry, and the periodicity is thus frequently imposed.
In one-dimensional dynamics this is usually the case.
According to the features of the particle system, sometimes it is require that the nonlinearity has spatial periodicity, which is not the case here since our~$F$ has no explicit dependence on the space variable~$x$.
Here we are in a similar setting as in~\cite{DingLiu2014periodic}, namely we seek for periodic solutions for the nonlinear Dirac equation~\eqref{eq:NDE-spatial}.
The difference to ~\cite{DingLiu2014periodic} lies in the nonlinearity, which will become clear later.

Geometrically, this amounts to consider the equation~\eqref{eq:NDE-spatial} on the compact manifold~$\mathbb{T}^3=\faktor{\mathbb{R}^3}{\Gamma}$.
The spectral properties for the linearized operators on torus is in great contrast to the one on~$\mathbb{R}^3$, so is the analysis needed for the variational scheme.
For example, we cannot appeal to the Fourier space to obtain enough linking structures, and we do not have a global Pohozaev identity to control the kinetic energy and the potential energy separately, which are crucial for the proof in~\cite{EstebanSere1995stationary} to obtain solutions on~$\mathbb{R}^3$.
On the other hand, the torus is still good enough to allow for variational analysis and we can changle local strategies to carry out the variational method in~\cite{EstebanSere1995stationary} to obtain nontrivial solutions.

The main motivation of the nonlinearities comes from the following examples.
In the Soler model~\cite{Soler1970Classical}, the self-coupling has the form
\begin{align}\label{eq:F Soler}
 F(\psi)
 =\frac{1}{2}G(\bar{\psi}\psi)
\end{align}
with~$G\in C^2(\mathbb{R},\R_+)$, and~$G(s)=0$ for~$s\leq 0$.
To solve the corresponding nonlinear Dirac equation on~$\mathbb{R}^3$, a suitable “radial” Ansatz were used to reduce the equation to ODEs and then the shooting method can provide some solutions.
These methods cannot be applied in the periodic setting for the lack of a good radail Ansatz.
A more general form of~$F$ arise in~\cite{Finkelstein1951Non-linea} where the physicists were studying the symmetric coupling between nucleons, muons, and leptons, for which
\begin{align}\label{eq:nonlinearity Finkelstein}
 F(\psi) = \frac{1}{2}|\bar{\psi}\psi|^2 + b|\bar{\psi}\gamma^5\psi|^2
\end{align}
where~$\gamma^5= \gamma^0\gamma^1\gamma^2\gamma^3$.
In such models, the existence theory remains still largely open.
Indeed, for~\eqref{eq:nonlinearity Finkelstein}, it has growth order~$|\psi|^4$ which is super-critical for the Sobolev embeddings, thus not appropriate for the variational scheme.

\item[(F5)] $\exists\quad\beta > 3$, $\forall\quad\delta > 0$, $\exists \quad C_\delta > 0$, $\forall \psi \in \mathbb{C}^4$, such that
\begin{align}
 |\nabla F(\psi)| \leq \left( \delta + C_\delta F(\psi)^{\frac{1}{\beta}} \right) |\psi|.
\end{align}
\end{itemize}
The practical nonlinearity that we aim to model include the Soler nonlinearity in~\eqref{eq:F Soler} as well as
\begin{align}
 F(\psi)= |\bar{\psi}\psi|^\alpha+ b|\bar{\psi}\gamma^5\psi|^\beta,
\end{align}
where~$\gamma^5=\gamma^0\gamma^1\gamma^2\gamma^3$ and~$b\geq 0$,~$1<\alpha,\beta<\frac{3}{2}$.

The main result of this work is the following.

\begin{align}\label{eq:4D nonlinear Dirac}
    \sum_{\mu=0}^{3} i\gamma^\mu\p_\mu\Psi-m \Psi + \gamma^0 \nabla F(\Psi)=0
\end{align}

\begin{align}\label{eq:F Soler}
    F(\psi)
    =\frac{1}{2}G(\bar{\psi}\psi)
\end{align}

\begin{align}\label{eq:NDE-spatial}\tag{*}
    -i\gamma^0 \gamma^k\p_k \psi + m\gamma^0\psi -a\psi-\nabla F(\psi)=0.
\end{align}

\begin{align}\label{eq:nonlinearity Finkelstein}
    F(\psi) = \frac{1}{2}|\bar{\psi}\psi|^2 + b|\bar{\psi}\gamma^5\psi|^2
\end{align}

An typical ansatz for the solutions are the so-called \emph{stationary solutions} or~\emph{solitary solutions} (see e.g.~\cite{EstebanSere1995stationary}) which takes the form
\begin{align}
 \Psi(t,\vec{x})= e^{-i a t}\psi(\vec{x})
\end{align}
Such solutions are regarded as the “particle-like solutions” and hence attract much attention.
For spinors of such form and for~$F$ invariant under rotations ( i.e. ~$F(e^{i\theta}\Psi)=F(\Psi)$),~\eqref{eq:4D nonlinear Dirac} is transformed into
\begin{align}
 i\gamma^k\p_k\psi + a\gamma^0\psi-m\psi +\gamma^0\nabla F(\psi)=0.
\end{align}
Equivalently, we are led to consider the equation
\begin{align}\label{eq:NDE-spatial}\tag{*}
 -i\gamma^0 \gamma^k\p_k \psi + m\gamma^0\psi -a\psi-\nabla F(\psi)=0.
\end{align}
Equations of this form have been studied in depth in literature, see e.g.~\cite{BalabaneCazenave1988Existence, Balabane1988Existence, Balabane1990Existence, Cazenave1986Existence,EstebanSere1995stationary,Merle1988Existence} and the references quoted therein.
Different from~\cite{EstebanSere1995stationary}, here we are concerned with periodic solutions, namely solutions satisfying
\begin{align}
 \psi(\theta+\tau)=\psi(\theta), \quad \forall \tau\in \Gamma
\end{align}
where~$\Gamma\subset \R^3$ is a 3-dimensional lattice.
They are referred as the standing periodic waves for~\eqref{eq:4D nonlinear Dirac}.

In quantum fields theory it is often required that the Hamiltonian has some symmetry, and the periodicity is thus frequently imposed.
In one-dimensional dynamics this is usually the case.
According to the features of the particle system, sometimes it is require that the nonlinearity has spatial periodicity, which is not the case here since our~$F$ has no explicit dependence on the space variable~$x$.
Here we are in a similar setting as in~\cite{DingLiu2014periodic}, namely we seek for periodic solutions for the nonlinear Dirac equation~\eqref{eq:NDE-spatial}.
The difference to ~\cite{DingLiu2014periodic} lies in the nonlinearity, which will become clear later.

Geometrically, this amounts to consider the equation~\eqref{eq:NDE-spatial} on the compact manifold~$\mathbb{T}^3=\faktor{\R^3}{\Gamma}$.
The spectral properties for the linearized operators on torus is in great contrast to the one on~$\R^3$, so is the analysis needed for the variational scheme.
For example, we cannot appeal to the Fourier space to obtain enough linking structures, and we do not have a global Pohozaev identity to control the kinetic energy and the potential energy separately, which are crucial for the proof in~\cite{EstebanSere1995stationary} to obtain solutions on~$\R^3$.
On the other hand, the torus is still good enough to allow for variational analysis and we can changle local strategies to carry out the variational method in~\cite{EstebanSere1995stationary} to obtain nontrivial solutions.

\item[(F5)] $\exists\quad\beta > 3$, $\forall\quad\delta > 0$, $\exists \quad C_\delta > 0$, $\forall \psi \in \mathbb{C}^4$, such that
\begin{align}
 |\nabla F(\psi)| \leq \left( \delta + C_\delta F(\psi)^{\frac{1}{\beta}} \right) |\psi|.
\end{align}
\end{itemize}
The practical nonlinearity that we aim to model include the Soler nonlinearity in~\eqref{eq:F Soler} as well as
\begin{align}
 F(\psi)= |\bar{\psi}\psi|^\alpha+ b|\bar{\psi}\gamma^5\psi|^\beta,
\end{align}
where~$\gamma^5=\gamma^0\gamma^1\gamma^2\gamma^3$ and~$b\geq 0$,~$1<\alpha,\beta<\frac{3}{2}$.

We outline the proof here.
The equation~\eqref{eq:NDE-spatial} admits a variational structure, namely it is the Euler--Lagrange equation of an action functional~$J$, see Section~\ref{sect:variational str}.
Unfortunately this functional~$J$ doesn't satisfy the Palais-Smale condition.
As a remedy we introduce a coercive perturbation multiplied by a small~$\eps\in (0,1]$ in Section~\ref{sect:perturbation}.
Then we show that the perturbed functional meets the Palais-Smale condition, although the~$(PS)_c$ sequences have a bound depending on~$\frac{1}{\eps}$.
Crucially using the spectral information of the Dirac operator, we will see that these perturbed functionals admits a linking structure in levels, which turn out to be uniform, i.e. with level estimates being uniform in~$\eps$.
Though the linking sets are both infinite dimensional, we use the negative gradient flow of the functional together with the Leray-Schauder degree to find a nontrivial linking.
Thus we obtain min-max solutions for the perturbed equations in the end of Section~\ref{sect:perturbation}.
By using the special structure of~$F$ and the perturbation term, we employ a contradiction argument to get uniform estimates for the perturbed solutions in Section~\ref{sect:uniform estimate}; the triviality of the kernel of the geometric Dirac operator on the three-sphere is needed.
This allows to pass along a subsequence to a nontrivial limit which is the desired nontrivial periodic solution, see Section~\ref{sect:last}.

\begin{prop}\label{prop:external}
 Assume that (F1-5) and~\eqref{eq:small oscillation} hold true, then~\eqref{eq:NLD with external field} admits a nontrivial periodic solution which is~$C^1$.
\end{prop}
Note that this differs from the results in~\cite{DingRuf2008Solutions} not only in the non-coercivity of the nonlinearity, but also in the growth order of the nonlinearity.
They dealt essential with the quadratic growth case, and we are concerned with the super-quadratic case, hence the results are not overlapping.
Furthermore, the multiplicity results are still missing in our case.
We will comment on this later, see Remark~\ref{rmk:multiplicity}.

The equation~\eqref{eq:NDE-spatial} is the Euler-Lagrange equation of the functional
\begin{align}
 J\colon H^{\frac{1}{2}}(M,\C^4)\to \R
\end{align}
given by
\begin{align}
 J(\psi)
 =\int_M \frac{1}{2}\Abracket{\psi,\pD\psi} -\frac{a}{2}|\psi|^2 - F(\psi)\dv.
\end{align}
Recall that the nonlinearity~$F$ satisfies the hypotheses (F1-5).
In particular,~$F(\psi)$ fails to be coercive in~$\psi$ in general.
Actually, in the model example,~~$F(\psi)=\frac{1}{2}G(\bar{\psi}\psi)$,
using the notation from~\eqref{eq:up-low}, we have
\begin{align}
 \bar{\psi}\psi = \Abracket{\gamma^0\psi,\psi}= |\psi_{\up}|^2 - |\psi_{\low}|^2, & &
 |\psi|^2 = |\psi_{\up}|^2 + |\psi_{\low}|^2 = \sum_{k=1}^4 |\psi^k|^2.
\end{align}
Note that~$\bar{\psi}\psi$ may be small, while~$|\psi|^2$ is large.
This kind of nonlinearity frequently arises in various particle models in quantum field theory, which is more complicated than the nonlinearities of the form~$G(|\psi|^2)$.
The approach we take here is motivated from~\cite{Bartsch1999nonlinear, EstebanSere1995stationary, HoferWysocki1990first,Sere1995homoclinic}.
Aside from the strongly indefinite nature of Dirac operators, there are new challenges in the periodic setting, such as the construction of the min-max structure, and the loss of radial solutions.
Methodologically, the powerful Fourier transform on~$\R^3$, which provides rich variational information in~\cite{EstebanSere1995stationary}, is no longer available.
Though one can still appeal to the Fourier series expansion of the spinors on~$\mathbb{T}^3$, we struggle to find a suitable linking structure for this problem which relies heavily on the spectral analysis of~$\pD$.
We aim to fix these issues and look for new variational periodic solutions.

Now we see that for each~$\eps\in (0,1]$ the perturbed functional~$J_\eps$ admits a min-max critical point~$\psi_\eps \in H^{\frac{1}{2}}$ such that
\begin{align}
 J_\eps(\psi_\eps) \in (c_1, c_2), & &
 \|\psi_\eps\|_{H^1}\leq C<+\infty.
\end{align}
Up to a subsequence, we assume that~$\psi_\eps\to \psi_0$ weakly in~$H^1$ and strongly in~$H^{\frac{1}{2}}\cap L^6$.
It follows that the limit spinor~$\psi_0$ solves the equation~\eqref{eq:NDE-spatial} in~$H^{-\frac{1}{2}}$, and it is nontrivial since
\begin{align}
 J(\psi_0)=\lim_{\eps\to 0^+} J_\eps(\psi_\eps) \geq c_1 >0.
\end{align}
 This is the desired nontrivial solution.

The regularity of the solution can be improved if~$F$ is smooth.

We outline the proof here.
The equation~\eqref{eq:NDE-spatial} admits a variational structure, namely it is the Euler--Lagrange equation of an action functional~$J$, see Section~\ref{sect:variational str}.
Unfortunately this functional~$J$ doesn't satisfy the Palais-Smale condition.
As a remedy we introduce a coercive perturbation multiplied by a small~$\varepsilon\in (0,1]$ in Section~\ref{sect:perturbation}.
Then we show that the perturbed functional meets the Palais-Smale condition, although the~$(PS)_c$ sequences have a bound depending on~$\frac{1}{\varepsilon}$.
Crucially using the spectral information of the Dirac operator, we will see that these perturbed functionals admits a linking structure in levels, which turn out to be uniform, i.e. with level estimates being uniform in~$\varepsilon$.
Though the linking sets are both infinite dimensional, we use the negative gradient flow of the functional together with the Leray-Schauder degree to find a nontrivial linking.
Thus we obtain min-max solutions for the perturbed equations in the end of Section~\ref{sect:perturbation}.
By using the special structure of~$F$ and the perturbation term, we employ a contradiction argument to get uniform estimates for the perturbed solutions in Section~\ref{sect:uniform estimate}; the triviality of the kernel of the geometric Dirac operator on the three-sphere is needed.
This allows to pass along a subsequence to a nontrivial limit which is the desired nontrivial periodic solution, see Section~\ref{sect:last}.

We also remark that, when there is an external field, the following equation is also considered:
\begin{align}\label{eq:NLD with external field}
    -i\gamma^0\gamma^k\p_k\psi + m\gamma^0\psi -a\psi - M\psi -\nabla F(x, \psi)=0,
\end{align}
where~$M$ stands for the external field, see~\cite{DingRuf2008Solutions}, which can be either vector or scalar.
If~$M(x)$ is a~$C^2$ continuous and positive, and does not oscillate much in the sense that
\begin{align}\label{eq:small oscillation}
    0< a \leq a + M < m,
\end{align}
then the argument carries over to this case, and we also obtain a nontrivial periodic~$C^1$ solution.

\begin{prop}\label{prop:external}
    Assume that (F1-5) and~\eqref{eq:small oscillation} hold true, then~\eqref{eq:NLD with external field} admits a nontrivial periodic solution which is~$C^1$.
\end{prop}
Note that this differs from the results in~\cite{DingRuf2008Solutions} not only in the non-coercivity of the nonlinearity, but also in the growth order of the nonlinearity.
They dealt essential with the quadratic growth case, and we are concerned with the super-quadratic case, hence the results are not overlapping.
Furthermore, the multiplicity results are still missing in our case.
We will comment on this later, see Remark~\ref{rmk:multiplicity}.

\

\noindent{\bf Acknowledgement.} We would like to thank Nadine Grosse and Meng Long for helpful conversation on these problems.

We outline the proof here. The equation~\eqref{eq:NDE-spatial} admits a variational structure, namely it is the Euler--Lagrange equation of an action functional~$J$ (see Section~\ref{sect:variational str}), but $J$ doesn't satisfy the Palais-Smale condition. As a remedy we introduce a coercive perturbation multiplied by a small~$\varepsilon\in (0,1]$ (Section~\ref{sect:perturbation}) and show the perturbed functional meets the Palais--Smale condition, although the $(PS)_c$ sequences have a bound depending on~$\frac{1}{\varepsilon}$. Using spectral information of the Dirac operator, the perturbed functionals admit a linking structure in levels with estimates uniform in~$\varepsilon$; although the linking sets are infinite dimensional, one uses the negative gradient flow together with the Leray--Schauder degree to find a nontrivial linking, yielding min-max solutions for the perturbed equations (end of Section~\ref{sect:perturbation}). Then, using the special structure of~$F$ and the perturbation term, a contradiction argument gives uniform estimates for the perturbed solutions (Section~\ref{sect:uniform estimate}); here the triviality of the kernel of the geometric Dirac operator on the three-sphere is needed. Finally, one passes along a subsequence to a nontrivial limit which is the desired nontrivial periodic solution (Section~\ref{sect:last}).

Assume that (F1-5) hold true.
Then there exists a nontrivial periodic solution for~\begin{align}\label{eq:NDE-spatial}\tag{*}
    -i\gamma^0 \gamma^k\p_k \psi + m\gamma^0\psi -a\psi-\nabla F(\psi)=0.
\end{align}, which is in~$C^1$.

Question:
Let \(\Gamma\subset \mathbb{R}^3\) be a 3-dimensional lattice, and call a spinor \(\psi:\mathbb{R}^3\to\mathbb{C}^4\) periodic if \(\psi(x+\tau)=\psi(x)\) for every \(\tau\in\Gamma\) (equivalently, \(\psi\) is defined on the torus \(\mathbb{T}^3=\mathbb{R}^3/\Gamma\)). Assume the nonlinearity \(F:\mathbb{C}^4\to\mathbb{R}\) satisfies hypotheses \((F1)\)–\((F5)\), and consider the nonlinear Dirac equation
\[
-i\gamma^0\gamma^k\partial_k\psi + m\gamma^0\psi - a\psi - \nabla F(\psi)=0
\]
(with summation over \(k=1,2,3\)). Here \(\gamma^\mu\) are the Dirac matrices. Which existence statement holds under these assumptions?

There exists a nontrivial \(\Gamma\)-periodic solution \(\psi\in C^1(\mathbb{R}^3,\mathbb{C}^4)\) (equivalently, a nonzero \(C^1\) spinor on \(\mathbb{T}^3\)) such that
\[
-i\gamma^0\gamma^k\partial_k\psi + m\gamma^0\psi - a\psi - \nabla F(\psi)=0.
\]

Choices:
B.
There exists a nontrivial \(\Gamma\)-periodic solution \(\psi\in C^1(\mathbb{R}^3,\mathbb{C}^4)\) such that
\[
-i\gamma^0\gamma^k\partial_k\psi + m\gamma^0\psi - a\psi - \nabla F(\psi)=0,
\]
and, moreover, this periodic solution exists for every choice of the parameter \(a\in\mathbb{R}\).
C.
There exists a nontrivial \(\Gamma\)-periodic weak solution \(\psi\) of
\[
-i\gamma^0\gamma^k\partial_k\psi + m\gamma^0\psi - a\psi - \nabla F(\psi)=0
\]
on \(\mathbb{T}^3\).
D.
For every 3-dimensional lattice \(\Gamma\subset\mathbb{R}^3\), there exists a nontrivial \(\Gamma\)-periodic solution \(\psi\in C^1(\mathbb{R}^3,\mathbb{C}^4)\) such that
\[
-i\gamma^0\gamma^k\partial_k\psi + m\gamma^0\psi - a\psi - \nabla F(\psi)=0.
\]
E.
There exists a nontrivial \(\Gamma\)-periodic solution \(\psi\in C^1(\mathbb{R}^3,\mathbb{C}^4)\) such that
\[
-i\gamma^0\gamma^k\partial_k\psi + m\gamma^0\psi - a\psi + \nabla F(\psi)=0.
\]

The stem does not state or strongly hint at the exact conclusion; it only gives the assumptions and asks which existence claim is valid.

The correct option is essentially the theorem statement itself: existence of a nontrivial Γ-periodic C1 solution under (F1)–(F5). This makes the item largely a recall/restatement question.

There is some reasoning in comparing stronger, weaker, and tampered variants (quantifiers, regularity, sign), but the item mainly tests recognition of the exact theorem rather than deriving the conclusion from mathematical structure.

The distractors are plausible and distinct: overstrong quantification in a or Γ, a weaker regularity conclusion, and a sign error in the equation. These align with common theorem-recall and statement-parsing mistakes.

Low answer leakage and strong distractors, but the question is mostly a theorem-statement recognition item rather than a genuinely generative reasoning task.


\label{main blocks}
The category $\mathcal{C}(\mathfrak{g},e_{sr},l,q)$ decomposes into blocks which are either trivial
(that is equivalent to the category of vector spaces) or of type $\Gamma$. In particular,
the category $\mathcal{C}(\mathfrak{g},e_{sr},l,q)$ is of tame representation type. Also the cohomology of
$\mathcal{C}(\mathfrak{g},e_{sr},l,q)$ (that is the Ext algebra of the unit object ${\bf 1}$) is isomorphic to the algebra
of invariants $S^\bullet(V)^\Gamma$.

\begin{theorem}\label{main 7mod}
(1) The category $\bar \C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\tilde D_4$.
In particular, it has 17 simple objects.

(2) We have $\FP(\bar \C(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\approx 3054.068811$.

(3) The category $\bar \C(G_2,G_2(a_1),7,q)$ has stable Chevalley property.

(4) The category $\bar \C(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial
commutative exact algebras.
\end{theorem}

\label{Witt}
In this section $\K$ is an algebraically closed field of arbitrary characteristic.
\subsection{Exact algebras} Let $\C$ be a finite tensor category over $\K$ and let $A=(A,m,u)\in \C$ be

\begin{theorem}\label{main 7}
(1) The category $\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\tilde E_7$.
In particular it has 23 simple objects.

(2) We have $\FP(\C(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\approx 18324.416384$.

(3) The category $\C(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple
objects are direct sums of simples and projectives.

(4) The M\"uger center of the category $\C(G_2,G_2(a_1),7,q)$ is equivalent to $\Rep(S_3)$
(where $S_3$ is the symmetric group on three letters).
\end{theorem}

\label{genconj}
\subsection{Categories $\C(G_2,G_2(a_1),l,q)$}
We expect that Theorem \ref{main 7} extends to the case of arbitrary undivisible (i.e. not
divisible by 3) $l\ge 7$.

\begin{conjecture}\

\begin{definition} \label{divisible}
We say that $l\in \Z_{\ge 1}$ is \textit{divisible} if $l$ is divisible
by $m$ and $l$ is \textit{undivisible} otherwise.     \end{definition}

\begin{conjecture}\label{g2conjm}
    Assume $l\ge 12$ and $l$ is divisible by 3. Then

    $$\FP(\C(G_2, G_2(a_1),l,q))=\frac29\frac{l^4}{(2\sin(\pi/l))^4(2\sin(2\pi/l))^2(2\sin(3\pi/l))^4}.$$
and for $l\ge 15$
$$\FP(T(\omega_1))=[3]_l+[5]_l-1,\; \FP(T(\omega_2))=2[7]_l-[5]_l+[3]_l+2.$$
\end{conjecture}

We checked that the formulas work for $l=12,15,18,21$; this applies also for $l=9$ if we
define category $\C(G_2, G_2(a_1),9,q)$ to be a semisimplification of $\T(\g,q)$
(for $q$ such that $q^2$ has order 9), see \cite{RW}. The formula for $\FP(T(\omega_2))$ gives
an incorrect result for $l=12$, namely the right hand side and the left hand side
differ exactly by 1. This is likely to be explained by the fact that $T(\omega_2)$
does not coincide with Weyl module in this (and only in this) case.

\subsection{Categories $\C(\g,e,l,q)$ for distinguished nilpotent $e\in \g$}\label{distconj}
Let $Q=Q(e)$ be the centralizer of a $sl_2-$triple associated with $e$ in the simply
connected group $G$ with $\mbox{Lie}(G)=\g$. In the case of distinguished nilpotent $e$,
$Q$ is a finite group. Let $2k_i+1, i\in M=M(e)$ be the sizes of Jordan cells for
the adjoint action of $e$ on $\g$ (it is well known that these sizes are odd for distinguished
nilpotent elements; the number of these cells, that is the cardinality of set $M$, is
the dimension of the centralizer of $e$ in $\g$). For any $k\in \Z_{>0}$ we set
$$S_k(l):=\frac{l}{(2^k\sin(\pi/l)\sin(2\pi/l)\cdots \sin(k\pi/l))^2}.$$

\begin{conjecture}\label{FPdim}
    Assume $m=1$ or $l$ is undivisible. Then
    $$\FP(\C(\g,e,l,q))=|Q|\prod_{i=1}^MS_{k_i}(l).$$
\end{conjecture}

\subsection{}
This paper is a contribution to the theory of braided finite tensor categories. In the case
of semisimple categories over $\mathbb{C}$ many known examples arise from Wess-Zumino-Witten
models in conformal field theory, see e.g. \cite{BK}. One of the easiest algebraic constructions of these categories was given
by H.~H.~Andersen \cite{An} (see also \cite{Saw}). In modern terms this construction can be summarized
as follows: for a simple Lie algebra $\mathfrak{g}$ and a root of unity $q$ such that $l$ is the order of $q^2$ one considers
the category $\mathcal{T}(\mathfrak{g},q)$ of {\em tilting modules} over the quantum group at a root of unity $q$ associated
with Lie algebra $\mathfrak{g}$. Then one defines a category $\mathcal{C}(\mathfrak{g}, l, q)$ as the {\em semisimplification}
of $\mathcal{T}(\mathfrak{g},q)$ (see e.g. \cite{EOsemi}). The category $\mathcal{C}(\mathfrak{g}, l, q)$ is a semisimple braided tensor
category; moreover this category is finite (i.e. it has only finitely many classes of simple objects) if
$l$ is sufficiently large. The precise bounds for $l$ are given in \cite[Figure 2]{Scho}; in this paper
we will always assume that $l$ is sufficiently large in this sense. Note that there is a combinatorial
difference between the case when $l$ is {\em divisible} (see \ref{divisible}) and when it is not.

The procedure of semisimplification above can be described as taking the quotient by a suitable
tensor ideal (namely, by the ideal of negligible morphisms). However, the category $\mathcal{T}(\mathfrak{g},q)$ has
many other tensor ideals. In \cite{CEO} for any {\em distinguished} nilpotent element $e\in \mathfrak{g}$ (or, in the case
when $l$ is divisible, $e\in \mathfrak{g}^L$ where $\mathfrak{g}^L$ is the Langlands dual Lie algebra of $\mathfrak{g}$),
a tensor ideal $\cI_e$ was constructed such that the quotient category $\mathcal{T}(\mathfrak{g},q)/\cI_e$ admits a {\em monoidal
abelian envelope} $\mathcal{C}(\mathfrak{g},e,l,q)$; moreover, the category $\mathcal{C}(\mathfrak{g},e,l,q)$ is a finite tensor category
in the sense of \cite{EOfinite}. For example when $e$ is a regular nilpotent element (which is always
distinguished), the ideal $\cI_e$ is the ideal of negligible morphisms and $\mathcal{C}(\mathfrak{g},e,l,q)=\mathcal{C}(\mathfrak{g},l,q)$.
Unfortunately, there is not much we can say about the category $\mathcal{C}(\mathfrak{g},e,l,q)$ for other nilpotent
elements $e$ (however, see Section \ref{distconj} for a conjectural formula for the Frobenius-Perron dimension
of $\mathcal{C}(\mathfrak{g},e,l,q)$ and conjectural description of its cohomology). The goal of this paper is to give
some explicit information about the category $\mathcal{C}(\mathfrak{g},e,l,q)$ in the case when $e=e_{sr}$ is a subregular
nilpotent element, see \cite[4.2]{CMg}. Recall that $e_{sr}$ is distinguished if and only if $\mathfrak{g}$ is not of
types $A_n, n\ge 1$ or $B_n, n\ge 2$, see \cite{CMg}.

To state our main results we need a bit more notation. Let $V$ be a two dimensional space and let
$\Gamma \subset SL(V)$ be a finite subgroup of even order. By the classical McKay correspondence,
the finite subgroups of $SL(V)$ up to conjugacy are labelled by simply laced affine Dynkin diagrams (the only
subgroups of odd order correspond to affine Dynkin diagrams of type $\tilde A_{2n}$). We associate
a subgroup $\Gamma$ to $\mathfrak{g}$ and $l$ as above as follows:
\renewcommand{\arraystretch}{1.4}
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\mathfrak{g}&B_n&C_n&D_n&E_n&F_4&G_2\\
&n\ge 3&n\ge 3&n\ge 4&n=6,7,8&&\\
\hline
l&\mbox{divisible}&\mbox{not divisible}&\mbox{any}&\mbox{any}&\mbox{any}&\mbox{any}\\
\hline
\Gamma &\tilde D_{2n} &\tilde D_{2n}&\tilde D_n&\tilde E_n&\tilde E_7&\tilde E_7\\
\hline
\end{array}
$$

Let $\wedge(V)$ be the exterior algebra of $V$. We can consider $\wedge(V)$ as an algebra in the category
$\mbox{Rep}(\Gamma)$ of finite dimensional representations of $\Gamma$. We consider the abelian
category of left $\wedge(V)-$modules in the category $\mbox{Rep}(\Gamma)$ and we call it
{\em block of type} $\Gamma$. Thus, a block of type $\Gamma$ is the category equivalent to that of finite dimensional
representations of the cross product of $\wedge(V)$ with the group algebra of $\Gamma$.
Finally let $S^\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \subset
S^\bullet(V)$ is in degree 2). The group $\Gamma$ acts on $S^\bullet(V)$ preserving the grading.
Our first main result describes the structure of abelian category $\mathcal{C}(\mathfrak{g},e_{sr},l,q)$.

Let $\wedge(V)$ be the exterior algebra of $V$. We can consider $\wedge(V)$ as an algebra in the category
$\mbox{Rep}(\Gamma)$ of finite dimensional representations of $\Gamma$. We consider the abelian
category of left $\wedge(V)-$modules in the category $\mbox{Rep}(\Gamma)$ and we call it
{\em block of type} $\Gamma$. Thus, a block of type $\Gamma$ is the category equivalent to that of finite dimensional
representations of the cross product of $\wedge(V)$ with the group algebra of $\Gamma$.
Finally let $S^\bullet(V)$ be the symmetric algebra of $V$ which is graded by even integers (so $V \subset
S^\bullet(V)$ is in degree 2). The group $\Gamma$ acts on $S^\bullet(V)$ preserving the grading.
Our first main result describes the structure of abelian category $\C(\g,e_{sr},l,q)$.

\begin{remark} The proof of Theorem \ref{main blocks} shows that the number of blocks of type $\Gamma$
is the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this
number is the same as the number of simple objects in the category $\C(\g,l,q)$.
\end{remark}

\begin{theorem}\label{main 7}
(1) The category $\C(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\tilde E_7$.
In particular it has 23 simple objects.

\begin{theorem}\label{main 7mod}
(1) The category $\bar \C(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\tilde D_4$.
In particular, it has 17 simple objects.

We consider the category $\Rep(\Gamma \ltimes V, \epsilon)$ of
finite dimensional representations of $\Gamma \ltimes V$ where $\epsilon$ acts as the parity automorphism,
see e.g. \cite[9.11]{EGNO}. Thus, $\Rep(\Gamma \ltimes V, \epsilon)$ is a symmetric finite tensor category.
As an abelian category it is equivalent to the category of representations of algebra
$\wedge(V)$ in the category $\Rep(\Gamma)$ (or, equivalently, representations of $\wedge(V)$ with an action
of $\Gamma$ compatible in an obvious sense). The simple objects of $\Rep(\Gamma \ltimes V, \epsilon)$ are irreducible representations of $\Gamma$ where $V\subset \wedge(V)$ acts by zero;
let $\Irr(\Gamma)$ be the set of isomorphism classes of such objects.
The projective cover of the unit object is $\wedge(V)$; and the projective cover of $V_i\in \Irr(\Gamma)$
is $V_i\otimes \wedge(V)$. In particular, the composition factors of the projective cover of $V_i$ are
$V_i$ appearing twice and the irreducible summands of $V_i\otimes V$. Thus, the Cartan matrix of
the category $\Rep(\Gamma \ltimes V, \epsilon)$ is $2\Id+A_\Gamma$ where $A_\Gamma$ is the adjacency
matrix of the McKay graph of $\Gamma$ (recall that the vertices of the McKay graph are elements
of $\Irr(\Gamma)$ and the number of edges between $V_i$ and $V_j$ is the multiplicity of $V_j$ as
a direct summand of $V_i\otimes V$).

\begin{corollary} \label{cor:self-dual}
 Let $\mathfrak g$ be of type $B_n, C_n, F_4, G_2, D_{2n}, E_7$, or $E_8$.
 Then all simple objects of $\C=\C(\mathfrak g, e_{sr}, l, q)$ are self-dual.
\end{corollary}
\begin{proof}
 It is well-known that in this case all simple representations of $\mathfrak g$ are self-dual. It follows from properties (1) and (2) in Section \ref{qgtitling} that the tilting modules $T(\lambda)$ are also self-dual.

\subsection{Proof of Theorem \ref{main blocks}} By Proposition \ref{Cartansr} the Cartan
matrix of the principal block of $\C(\g,e_{sr},l,q)$ is $2\Id+A(X_{sr})$ where $X_{sr}$ is
some affine Dynkin diagram not of type $A$, so it is a tree. Thus, by Proposition \ref{tree}
(applied to this block and to the category $\Rep(\Gamma \ltimes V)$)
we see that the principal block is equivalent to $\Rep(\Gamma \ltimes V)$ where
$V$ is as in \ref{toy} and $\Gamma \subset SL(V)$ is a finite subgroup with McKay graph $X_{sr}$.
Using the translation functors (which descend to the category $\T(\g,q)/\cI_{e_{sr}}$) we see
that all the other blocks of $\C(\g,e_{sr},l,q)$ involving $T(w\cdot \lambda)$ where $\lambda$
is in the interior of the fundamental alcove are equivalent to the principal block, and hence
to the category $\Rep(\Gamma \ltimes V)$. The other blocks involving $T(\lambda)$ with $\lambda$
on the wall are trivial by Corollaries \ref{nog222} and \ref{cor:on the wall}. The remaining statements of
Theorem \ref{main blocks} follow from Proposition \ref{3.1}.

It is a classical result of P.~Slodowy that the singularity of the nilpotent
cone of $\g$ at the point $e_{sr}$ is $\BC^2/\Gamma$ for a suitable subgroup
$\Gamma \subset SL_2(\BC)$, see \cite{Sl}. Comparing this with Theorem \ref{main blocks}
we expect that the cohomology of the category $\C(\g,e,l,q)$ is related
with the singularity of the nilpotent cone at point $e$. Thus let $\Sl_e$
be the Slodowy slice at the point $e$, see e.g. \cite{GG}. The variety $\Sl_e$
is equipped with an action of $Q\times \BC^\times$. Thus, the algebra of functions
$\mO(Sl_e)$ is graded and is equipped with a grading-preserving action of $Q$;
let $\mO(Sl_e)^Q\subset \mO(Sl_e)$ be the algebra of invariants.

\begin{theorem}\label{main blocks}
The category $\C(\g,e_{sr},l,q)$ decomposes into blocks which are either trivial
(that is equivalent to the category of vector spaces) or of type $\Gamma$. In particular,
the category $\C(\g,e_{sr},l,q)$ is of tame representation type. Also the cohomology of
$\C(\g,e_{sr},l,q)$ (that is the Ext algebra of the unit object $\be$) is isomorphic to the algebra
of invariants $S^\bullet(V)^\Gamma$.
\end{theorem}

\begin{remark} The proof of Theorem \ref{main blocks} shows that the number of blocks of type $\Gamma$
is the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this
number is the same as the number of simple objects in the category $\mathcal{C}(\mathfrak{g},l,q)$.
\end{remark}

\subsection{} Next, we study one specific example, the category $\mathcal{C}(G_2,G_2(a_1),7,q)$ (thus, we
consider Lie algebra $\mathfrak{g}$ of type $G_2$; also $G_2(a_1)$ is the standard notation for the subregular
nilpotent orbit in type $G_2$). This is the simplest example of the categories considered above
(at least for undivisible $l$). Recall the standard notation for the quantum numbers:
$$[k]_l=\frac{\sin(k\pi/l)}{\sin(\pi/l)}.$$
In particular, $[2]_7=[5]_7=2\cos(\pi/7)\approx 1.801938$ and $[3]_7=[4]_7=\frac{\sin(3\pi/7)}{\sin(\pi/7)}\approx
2.246980$.

\begin{theorem}\label{main 7}
(1) The category $\mathcal{C}(G_2,G_2(a_1),7,q)$ has 15 trivial blocks and one block of type $\tilde E_7$.
In particular it has 23 simple objects.

(2) We have $\text{FPdim}(\mathcal{C}(G_2,G_2(a_1),7,q))=294(7+15[3]_7+12[5]_7)\approx 18324.416384$.

(3) The category $\mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property: tensor products of simple
objects are direct sums of simples and projectives.

(4) The M\"uger center of the category $\mathcal{C}(G_2,G_2(a_1),7,q)$ is equivalent to $\text{Rep}(S_3)$
(where $S_3$ is the symmetric group on three letters).
\end{theorem}

In view of Theorem \ref{main 7} (4), it makes sense to consider the de-equivariantization $\bar \mathcal{C}(G_2,G_2(a_1),7,q)$
of $\mathcal{C}(G_2,G_2(a_1),7,q)$ with respect to its M\"uger center (so the category $\bar \mathcal{C}(G_2,G_2(a_1),7,q)$
is the Brugui\`eres' modularisation of $\mathcal{C}(G_2,G_2(a_1),7,q)$, see \cite{Brug}). The category
$\bar \mathcal{C}(G_2,G_2(a_1),7,q)$ is a non-semisimple modular tensor category in the sense of Shimizu,
see \cite{Shim}.

\begin{theorem}\label{main 7mod}
(1) The category $\bar \mathcal{C}(G_2,G_2(a_1),7,q)$ has 12 trivial blocks and one block of type $\tilde D_4$.
In particular, it has 17 simple objects.

(2) We have $\text{FPdim}(\bar \mathcal{C}(G_2,G_2(a_1),7,q))=49(7+15[3]_7+12[5]_7)\approx 3054.068811$.

(3) The category $\bar \mathcal{C}(G_2,G_2(a_1),7,q)$ has stable Chevalley property.

(4) The category $\bar \mathcal{C}(G_2,G_2(a_1),7,q)$ is completely anisotropic: it has no non-trivial
commutative exact algebras.
\end{theorem}

In \cite[Question 7.20]{ShYa} (see also \cite[Question 6.25]{LaWa}) K.~Shimizu and H.~Yadav asked whether non-semisimple completely
anisotropic categories exist;  Theorem \ref{main 7mod} (4) gives a positive answer to this question.

In \cite[Definition 6.23]{LaWa} (see also \cite[Definition 7.2]{ShYa}) R.~Laugwitz and C.~Walton defined an important Witt equivalence
relation on the set of non-degenerate braided finite tensor categories. In Section \ref{Witt} we prove
some general properties of this relation which imply

\begin{theorem}\label{main 7Witt}
The category $\bar \mathcal{C}(G_2,G_2(a_1),7,q)$ is not Witt equivalent to any semisimple category.
\end{theorem}

Thus, the non-semisimple Witt group is different from its semisimple version studied in \cite{DMNO}.

\subsection{} In Section \ref{genconj} we present some conjectures. Most importantly, we expect that the categories
$\mathcal{C}(\mathfrak{g},e,l,q)$  make sense for all nilpotent elements $e\in \mathfrak{g}$ (or $e\in \mathfrak{g}^L$
in the divisible case). If $e$ is not distinguished,
the category $\mathcal{C}(\mathfrak{g},e,l,q)$ is not finite; however we expect that it is obtained from a finite tensor
category by equivariantization.

\subsection{Acknowledgements} Some ideas that led to this paper were inspired by participation of one of us (V.O.) in American Institute of Mathematics SQuaRE ``Lie algebras in symmetric tensor categories''; we are very grateful to this institution and to the fellow participants Iv\'an Angiono, Agustina Czenky, Pavel Etingof, Julia Plavnik, and Guillermo Sanmarco. We also thank Dmitri Nikshych and Kenichi Shimizu for useful discussions.

The only proof-related information in the post-theorem text is the remark:

\begin{remark} The proof of Theorem \ref{main blocks} shows that the number of blocks of type $\Gamma$
is the same as the number of weights inside of the fundamental alcove (but not on its boundary). Thus, this
number is the same as the number of simple objects in the category $\mathcal{C}(\mathfrak{g},l,q)$.\end{remark}

No further outline/steps of the proof of Theorem~\ref{main blocks} are given.

Question: Let $\mathfrak g$ be a simple Lie algebra and let $q$ be a root of unity such that $l$ is the order of $q^2$, with $l$ sufficiently large. Assume that $(\mathfrak g,l)$ is one of the following cases: $\mathfrak g=B_n$ ($n\ge 3$) with $l$ divisible; $\mathfrak g=C_n$ ($n\ge 3$) with $l$ not divisible; $\mathfrak g=D_n$ ($n\ge 4$) with any $l$; $\mathfrak g=E_n$ ($n=6,7,8$) with any $l$; $\mathfrak g=F_4$ with any $l$; or $\mathfrak g=G_2$ with any $l$. Let $e_{sr}$ be the subregular nilpotent element, and let $\mathcal C(\mathfrak g,e_{sr},l,q)$ be the corresponding finite tensor category. Let $V$ be a $2$-dimensional vector space, and let $\Gamma\subset SL(V)$ be the finite subgroup attached to $(\mathfrak g,l)$ by the McKay correspondence, namely of affine Dynkin type $\tilde D_{2n}$ for $B_n$ and $C_n$, $\tilde D_n$ for $D_n$, $\tilde E_n$ for $E_n$, and $\tilde E_7$ for $F_4$ and $G_2$. A block of type $\Gamma$ means the abelian category of left $\wedge(V)$-modules in $\operatorname{Rep}(\Gamma)$, equivalently the category of finite-dimensional representations of the crossed product algebra $\wedge(V)\rtimes \Gamma$. If $S^\bullet(V)$ denotes the symmetric algebra of $V$ (with the natural $\Gamma$-action), which statement holds for every such category $\mathcal C(\mathfrak g,e_{sr},l,q)$?

The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\Gamma$. Consequently $\mathcal C(\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\operatorname{Ext}^\bullet_{\mathcal C}(\mathbf 1,\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\bullet(V)^\Gamma$.

Choices:
B. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is of type $\Gamma$. Consequently $\mathcal C(\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\operatorname{Ext}^\bullet_{\mathcal C}(\mathbf 1,\mathbf 1)$ of the unit object, is isomorphic to the full symmetric algebra $S^\bullet(V)$.
C. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\Gamma$. In particular, $\mathcal C(\mathfrak g,e_{sr},l,q)$ has tame representation type.
D. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\Gamma$. Consequently $\mathcal C(\mathfrak g,e_{sr},l,q)$ has finite representation type, and its cohomology, i.e. the Ext algebra $\operatorname{Ext}^\bullet_{\mathcal C}(\mathbf 1,\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\bullet(V)^\Gamma$.
E. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or Morita equivalent to $\operatorname{Rep}(\Gamma)$. Consequently $\mathcal C(\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\operatorname{Ext}^\bullet_{\mathcal C}(\mathbf 1,\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\bullet(V)^\Gamma$.

The stem defines the setup and the meaning of a block of type Γ, but it does not explicitly reveal the correct decomposition, representation type, or cohomology algebra.

The correct option is essentially a direct statement of the target theorem/result: block decomposition, tame type, and the Ext-algebra identification are all restated almost verbatim.

The item requires some discrimination among closely related statements (tame vs finite type, invariants vs full symmetric algebra, global vs blockwise cohomology), but it mostly tests theorem recall rather than substantial mathematical generation.

The distractors are plausible and mathematically targeted: they reflect realistic errors such as strengthening to finite type, omitting trivial blocks, or confusing global and blockwise cohomology.

A solid recall-based MCQ with strong distractors and little answer leakage, but it is largely a theorem restatement and does not strongly test generative reasoning.


Question: Let $\mathfrak g$ be a simple Lie algebra and let $q$ be a root of unity such that $l$ is the order of $q^2$, with $l$ sufficiently large. Assume that $(\mathfrak g,l)$ is one of the following cases: $\mathfrak g=B_n$ ($n\ge 3$) with $l$ divisible; $\mathfrak g=C_n$ ($n\ge 3$) with $l$ not divisible; $\mathfrak g=D_n$ ($n\ge 4$) with any $l$; $\mathfrak g=E_n$ ($n=6,7,8$) with any $l$; $\mathfrak g=F_4$ with any $l$; or $\mathfrak g=G_2$ with any $l$. Let $e_{sr}$ be the subregular nilpotent element, and let $\mathcal C(\mathfrak g,e_{sr},l,q)$ be the corresponding finite tensor category. Let $V$ be a $2$-dimensional vector space, and let $\Gamma\subset SL(V)$ be the finite subgroup attached to $(\mathfrak g,l)$ by the McKay correspondence, namely of affine Dynkin type $\tilde D_{2n}$ for $B_n$ and $C_n$, $\tilde D_n$ for $D_n$, $\tilde E_n$ for $E_n$, and $\tilde E_7$ for $F_4$ and $G_2$. A block of type $\Gamma$ means the abelian category of left $\wedge(V)$-modules in $\operatorname{Rep}(\Gamma)$, equivalently the category of finite-dimensional representations of the crossed product algebra $\wedge(V)\rtimes \Gamma$. If $S^\bullet(V)$ denotes the symmetric algebra of $V$ (with the natural $\Gamma$-action), which statement holds for every such category $\mathcal C(\mathfrak g,e_{sr},l,q)$?

Choices:
B. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is of type $\Gamma$. Consequently $\mathcal C(\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\operatorname{Ext}^\bullet_{\mathcal C}(\mathbf 1,\mathbf 1)$ of the unit object, is isomorphic to the full symmetric algebra $S^\bullet(V)$.
C. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\Gamma$. In particular, $\mathcal C(\mathfrak g,e_{sr},l,q)$ has tame representation type.
D. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or of type $\Gamma$. Consequently $\mathcal C(\mathfrak g,e_{sr},l,q)$ has finite representation type, and its cohomology, i.e. the Ext algebra $\operatorname{Ext}^\bullet_{\mathcal C}(\mathbf 1,\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\bullet(V)^\Gamma$.
E. The category $\mathcal C(\mathfrak g,e_{sr},l,q)$ decomposes into blocks each of which is either trivial (equivalent to the category of vector spaces) or Morita equivalent to $\operatorname{Rep}(\Gamma)$. Consequently $\mathcal C(\mathfrak g,e_{sr},l,q)$ has tame representation type, and its cohomology, i.e. the Ext algebra $\operatorname{Ext}^\bullet_{\mathcal C}(\mathbf 1,\mathbf 1)$ of the unit object, is isomorphic to the invariant algebra $S^\bullet(V)^\Gamma$.

The stem does not explicitly state the correct conclusion. It supplies technical setup and definitions, but it does not directly reveal the decomposition, tame type, and cohomology identification.

This is close to a theorem-recall item: the stem gives the hypotheses of a specific result and asks for the matching conclusion. It is not a pure restatement because the options differ in subtle but meaningful ways, yet it remains largely theorem recognition.

Some reasoning is needed to distinguish between nearby statements (tame vs finite type, invariants vs full symmetric algebra, block type vs Morita equivalence, complete vs weaker true statement). However, it mainly tests precise recall/comparison rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted. They reflect common failure modes: overstrengthening, weakening, confusing invariants with the full algebra, and replacing the intended block description with a nearby categorical notion.

A solid MCQ with strong distractors and little answer leakage, but it is primarily a precise theorem-identification question rather than a genuinely generative reasoning task.


There are two important operations that one can perform on a finite volume hyperbolic 3-manifold. One can drill out a simple closed geodesic and one can Dehn fill a cusp. It is a well-known phenomenon that all drillings and most fillings give back a hyperbolizable 3-manifold whose volume is larger for the first operation and smaller for the second. The goal of this short note is to prove two results that give a {\em linear} quantitative control on how much the volume can change.

\subsection*{Drilling}
Let us start with drillings. We prove the following.

\subsection*{Drilling}
Let us start with drillings. We prove the following.

\begin{corA}
\label{cor:main}
There exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\gamma\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\gamma$ has volume
\[
{\rm vol}(M-\gamma)\le{\rm vol}(M)+c\ell
\]
where $\ell=2\cdot{\rm inj}(M)$ is the length of $\gamma$.
\end{corA}

Similar estimates on the volume increase under drilling operations on hyperbolic 3-manifolds were already considered by Bridgeman \cite{B}, Agol \cite{A}, Agol, Storm, Thurston, and Dunfield \cite[Theorem 10.1]{AST} where they proved the existence of explicit functions $f,g>1$ such that ${\rm vol}(M-\gamma)\le f(R){\rm vol}(M)+g(R)\ell$. So, the main achievement of this note is to get rid of the multiplicative constant in front of ${\rm vol}(M)$. This shows that the increase in volume only depends on the {\em local} geometry around $\gamma$.

Theorem \ref{thm:main} goes in the direction of a conjecture of Bridgeman \cite{B}, which asks whether there is a linear upper bound on the volume increase under drilling operations. While very good estimates are available when the drilled geodesics are short (see the work of Hodgson and Kerckhoff \cite{HK}), it is still unclear what happens for long geodesics. Note that as long as we can uniformly bound from below $R$ the increase in volume is at most linear. This leads immediately to Corollary \ref{cor:main}. In fact, it is well-known that a shortest geodesic in a closed hyperbolic 3-manifold has a tubular neighborhood of radius at least $\ell/4$ where $\ell$ is the length of the geodesic (see \cite[Proposition 1.11]{GMT}). Combined with standard consequences of Margulis Lemma, this provides a universal constant $R_0>0$ such that a shortest geodesic in a closed hyperbolic 3-manifold $M$ always has a tubular neighborhood of radius $R\ge R_0$ (for much more refined estimates see \cite[Theorem 4.1]{GMT}).

\begin{thmA}
\label{thm:main'}
There exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\partial C$. Consider a simple closed geodesic $\mu\subset\partial C$ for the intrinsic flat metric on $\partial C$. If the length of $\mu$ is $\ell>\sqrt{c}\pi$ and the area of $\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\mu$, denoted by $M_\mu$, has a hyperbolic metric of volume
\[
{\rm vol}(M_\mu)+\frac{A}{2}\frac{\pi^2}{\ell^2}\left(1-c\frac{\pi^2}{\ell^2}\right)\le{\rm vol}(M).
\]
\end{thmA}

To prove Theorem \ref{thm:main}, we follow the approach of \cite[Theorem 10.1]{AST} with a difference. We first change the metric of $M$ inside a tubular neighborhood $T$ of $\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp. We do so by keeping under control the scalar curvature (see Lemma \ref{lem:functiondrill}). Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\gamma$. Instead of relying on Perelman's monotonicity formula, we exploit some other consequences of Perelman's work \cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \cite{And} and Kleiner and Lott \cite{KL}. This allows us to keep track of the fact that we {\em did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase.

\begin{lem}
\label{lem:functionfill}
There exists a constant $c>0$ such that the following holds. For every $\ell_1>2\pi,\ell_2>0$ there exist functions $a_{\rm fill},b_{\rm fill}:\mb{R}\to\mb{R}$ that satisfy the following properties. Let $R=\log(\ell_1/\pi)$ and $\kappa=2\pi\ell_2/\ell_1$.
\begin{enumerate}
 \item{$a_{\rm fill},b_{\rm fill}>0$.}
 \item{$a_{\rm fill}(r)=2\pi\sinh(r+R),b(r+R)=\kappa\cosh(r+R)$ on $(-R,-2\delta)$.}
 \item{$a_{\rm fill}(r)=\ell_1e^r,b(r)=\ell_2 e^r$ on $(-\delta,0)$.}
 \item{We have
 \[
 \left|\frac{a_{\rm fill}''}{a_{\rm fill}}+\frac{b_{\rm fill}''}{b_{\rm fill}}+\frac{a_{\rm fill}'b_{\rm fill}'}{a_{\rm fill}b_{\rm fill}}-3\right|\le c\frac{\pi^4}{\ell_1^4}.
 \]
 }
 \item{We have
 \[
 a_{\rm fill}b_{\rm fill}\le\ell_1\ell_2e^{2r}.
 \]
 }
\end{enumerate}
\end{lem}

\begin{pro}
\label{pro:metric on tube}
There exists a constant $c>0$ such that the following holds. Let $V$ be a solid torus. Suppose that the boundary $\partial V$ is equipped with a flat metric of area $A$. Assume that the length of the flat geodesic representative on $\partial V$ of the meridian $\mu\subset\partial V$ is $\ell>2\pi$. Then there exists a smooth Riemannian metric $(V,g)$ with the following properties.
\begin{itemize}
 \item{The metric $g$ is a hyperbolic cusp metric in a collar of $\partial V$, the boundary $\partial V$ is a horospherical section of the cusp, and the restriction of $g$ to the boundary agrees with the prescribed flat metric on $\partial V$.}
 \item{Denote by $S_g$ the scalar curvature of the metric $g$. We have
 \[
 \int_V{\left(\frac{|S_g|}{6}\right)^{3/2}{\rm dvol}_g}\le\frac{A}{2}\left(1-\frac{\pi^2}{\ell^2}\right)\left(1+c\frac{\pi^4}{\ell^4}\right).
 \]
 }
\end{itemize}
\end{pro}

\begin{lem}
\label{lem:functiondrill}
There exists a constant $c>0$ such that the following holds. For every $R>0$ there exist functions $a_{\rm drill},b_{\rm drill}:\mb{R}\to\mb{R}$ such that
\begin{enumerate}
    \item{$a_{\rm drill},b_{\rm drill}>0$.}
    \item{$a_{\rm drill}(r)=\sinh(r),b_{\rm drill}(r)=\cosh(r)$ on $(2R/3,\infty)$.}
    \item{$a_{\rm drill}(r)=\sinh(R)e^r,\cosh(R)b_{\rm drill}(r)=e^r$ on $(-\infty,R/3)$.}
    \item{On $[R/3,2R/3]$ we have
    \[
    \left|\frac{a_{\rm drill}''}{a_{\rm drill}}+\frac{b_{\rm drill}''}{b_{\rm drill}}+\frac{a_{\rm drill}'b_{\rm drill}'}{a_{\rm drill}b_{\rm drill}}\right|\le c\frac{1}{R^2}.
    \]
    }
    \item{We have $a_{\rm drill}b_{\rm drill}\le \sinh(R)\cosh(R)e^{2r}$.}
\end{enumerate}
\end{lem}

\begin{thmA}
\label{thm:main}
There exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\gamma\subset M$ be a simple closed geodesic. Suppose that $\gamma$ has length $\ell$ and has an embedded tubular neighborhood of radius at least $R\in(0,1)$. Then the unique hyperbolic metric on $M-\gamma$ has volume
\[
{\rm vol}(M-\gamma)\le{\rm vol}(M)+c\frac{\ell}{R}.
\]
\end{thmA}

\begin{corA}
\label{cor:main}
There exists a constant $c>0$ such that the following holds. Let $M$ be a closed hyperbolic 3-manifold and let $\gamma\subset M$ be a shortest non-trivial closed geodesic in $M$. Then the unique hyperbolic metric on $M-\gamma$ has volume
\[
{\rm vol}(M-\gamma)\le{\rm vol}(M)+c\ell
\]
where $\ell=2\cdot{\rm inj}(M)$ is the length of $\gamma$.
\end{corA}

Similar estimates on the volume increase under drilling operations on hyperbolic 3-manifolds were already considered by Bridgeman \cite{B}, Agol \cite{A}, Agol, Storm, Thurston, and Dunfield \cite[Theorem 10.1]{AST} where they proved the existence of explicit functions $f,g>1$ such that ${\rm vol}(M-\gamma)\le f(R){\rm vol}(M)+g(R)\ell$. So, the main achievement of this note is to get rid of the multiplicative constant in front of ${\rm vol}(M)$. This shows that the increase in volume only depends on the {\em local} geometry around $\gamma$.

Theorem \ref{thm:main} goes in the direction of a conjecture of Bridgeman \cite{B}, which asks whether there is a linear upper bound on the volume increase under drilling operations. While very good estimates are available when the drilled geodesics are short (see the work of Hodgson and Kerckhoff \cite{HK}), it is still unclear what happens for long geodesics. Note that as long as we can uniformly bound from below $R$ the increase in volume is at most linear. This leads immediately to Corollary \ref{cor:main}. In fact, it is well-known that a shortest geodesic in a closed hyperbolic 3-manifold has a tubular neighborhood of radius at least $\ell/4$ where $\ell$ is the length of the geodesic (see \cite[Proposition 1.11]{GMT}). Combined with standard consequences of Margulis Lemma, this provides a universal constant $R_0>0$ such that a shortest geodesic in a closed hyperbolic 3-manifold $M$ always has a tubular neighborhood of radius $R\ge R_0$ (for much more refined estimates see \cite[Theorem 4.1]{GMT}).

Let us mention the fact that the restriction to simple geodesics in closed hyperbolic 3-manifolds, rather than dealing with geodesic links in arbitrary finite volume ones, is not important. A straightforward generalization of Theorem \ref{thm:main} holds in this larger setting with exactly the same arguments and more robust bookkeeping. We chose to restrict ourselves to the simple case to streamline the exposition.

\subsection*{Filling}
Now we move to fillings. We give a new proof of the following lower bound due to Hodgson and Kerckhoff \cite{HK} using completely different techniques.

\begin{thmA}
\label{thm:main'}
There exists a constant $c>0$ such that the following holds. Let $M$ be a finite volume hyperbolic 3-manifold with one cusp. Let $C$ be a cusp neighborhood of $M$ bounded by the horospherical torus $\partial C$. Consider a simple closed geodesic $\mu\subset\partial C$ for the intrinsic flat metric on $\partial C$. If the length of $\mu$ is $\ell>\sqrt{c}\pi$ and the area of $\partial C$ is $A$ then the Dehn filling of $M-C$ with slope $\mu$, denoted by $M_\mu$, has a hyperbolic metric of volume
\[
{\rm vol}(M_\mu)+\frac{A}{2}\frac{\pi^2}{\ell^2}\left(1-c\frac{\pi^2}{\ell^2}\right)\le{\rm vol}(M).
\]
\end{thmA}

We remark that our assumptions differ slightly from the ones of \cite{HK} as we ask that $\ell\ge\sqrt{c}\pi$ whereas \cite{HK} requires that $\ell^2/A$ is bigger than a universal constant.

The problem of understanding how much the volume decreases under filling is better understood compared to the drilling case and has been studied with various methods. Thurston \cite[Chapter 6]{ThuNotes} showed that ${\rm vol}(M_\mu)<{\rm vol}(M)$ and that for every {\em fixed} $M$ we have ${\rm vol}(M_\mu)\to{\rm vol}(M)$ as $\ell\to\infty$. Neumann and Zagier \cite{NZ} then clarified, for every {\em fixed} $M$, the asymptotic behavior of ${\rm vol}(M)-{\rm vol}(M_\mu)$ as a function of $A/\ell^2$. Later, Hodgson and Kerckhoff \cite{HK}, gave a {\em universal} bound of the form ${\rm vol}(M)-{\rm vol}(M_\mu)= A\pi^2/2\ell^2+O(A^2/\ell^4)$.

The restriction to Dehn fillings of hyperbolic 3-manifolds of finite volume with one cusp rather than allowing multiple Dehn fillings of a selection of cusps of an arbitrary finite volume hyperbolic 3-manifold is not really important. A straightforward generalization of Theorem \ref{thm:main'} holds in that larger setting with the same proof but more notation. In order to keep the exposition as linear and short as possible, we restricted ourselves to the simple setup.

\subsection*{On the proofs}
Lastly, a few words on the proofs.

To prove Theorem \ref{thm:main}, we follow the approach of \cite[Theorem 10.1]{AST} with a difference. We first change the metric of $M$ inside a tubular neighborhood $T$ of $\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp. We do so by keeping under control the scalar curvature (see Lemma \ref{lem:functiondrill}). Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\gamma$. Instead of relying on Perelman's monotonicity formula, we exploit some other consequences of Perelman's work \cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \cite{And} and Kleiner and Lott \cite{KL}. This allows us to keep track of the fact that we {\em did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase.

The proof of Theorem \ref{thm:main'} follows exactly the same strategy. We change the metric of $M$ in a cusp neighborhood $C$ as in Gromov-Thurston's $2\pi$-Theorem \cite[Theorem 9]{BH} (of which we prove a quantitative version, see Proposition \ref{pro:metric on tube}). This time the interpolation between the metric on the cusp and the metric on a tube is more delicate as we want to keep under control the scalar curvature up to higher order (see Lemma \ref{lem:functionfill}). Then we use the sigma invariant comparison to conclude.

\subsection*{Acknowledgements}
I warmly thank Ian Agol and Martin Bridgeman for useful discussions and generous feedback on a first draft of this article.

To prove Theorem~\ref{thm:main}, the authors “follow the approach of \cite[Theorem 10.1]{AST} with a difference.” The steps described are:
\begin{itemize}
\item “We first change the metric of $M$ inside a tubular neighborhood $T$ of $\gamma$ interpolating between the metric on the tube and the metric on a suitable hyperbolic cusp.”
\item This is done “by keeping under control the scalar curvature (see Lemma~\ref{lem:functiondrill}).”
\item “Then we compare the volume of the modified metric with that of the hyperbolic structure on $M-\gamma$.”
\item “Instead of relying on Perelman's monotonicity formula,” they “exploit some other consequences of Perelman's work \cite{Pe1,Pe2}, namely, the exact computation of the sigma invariant of closed hyperbolic 3-manifolds as achieved by Anderson \cite{And} and Kleiner and Lott \cite{KL}.”
\item “This allows us to keep track of the fact that we \emph{did not change} the metric outside the tubular neighborhood $T$ and obtain an additive error rather than a multiplicative one in the volume increase.”
\end{itemize}

Existential–Universal
Inequality or Bound

Question: Let $M$ be a closed hyperbolic $3$-manifold, and let $\gamma\subset M$ be a simple closed geodesic of length $\ell$. Assume that $\gamma$ has an embedded tubular neighborhood of radius at least $R$, where $R\in(0,1)$. For the drilled manifold $M-\gamma$, equipped with its unique hyperbolic metric, which quantitative volume estimate holds?

There exists a constant $c>0$, independent of $M$, $\gamma$, $\ell$, and $R$, such that
\[
{\rm vol}(M-\gamma)\le {\rm vol}(M)+c\frac{\ell}{R}.
\]

Choices:
B.
There exists a constant $c>0$, independent of $M$, $\gamma$, $\ell$, and $R$, such that
\[
{\rm vol}(M-\gamma)\le {\rm vol}(M)+c\frac{\ell}{R^2}.
\]
C.
There exists a constant $c>0$, independent of $M$, $\gamma$, $\ell$, and $R$, such that
\[
{\rm vol}(M-\gamma)\le {\rm vol}(M)+c\ell.
\]
D.
For every $R\in(0,1)$ there exists a constant $c_R>0$, depending only on $R$, such that for every closed hyperbolic $3$-manifold $M$ and every simple closed geodesic $\gamma\subset M$ of length $\ell$ with an embedded tubular neighborhood of radius at least $R$,
\[
{\rm vol}(M-\gamma)\le {\rm vol}(M)+c_R\ell.
\]
E.
There exists a constant $c>0$, independent of $M$, $\gamma$, $\ell$, and $R$, such that
\[
{\rm vol}(M-\gamma)\le \left(1+c\frac{\ell}{R}\right){\rm vol}(M).
\]

linear dependence on inverse tube radius

dropped the factor $1/R$ using only that $R<1$

uniformity of the constant with respect to $R$

additive error versus multiplicative volume factor

The stem only defines terms and notation; it does not reveal the specific bound, the dependence on R, or the quantifier structure that distinguishes the correct choice.

Although the correct option is theorem-like, the question is not a bare restatement: the choices differ by domain restrictions on R, quantifier order, and the error-term dependence, so the student must discriminate among competing formulations.

The item requires some reasoning about how changing R-ranges, swapping quantifiers, or replacing ell/R by ell or ell R affects validity. However, it still leans more toward precise theorem recognition than toward substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: they target common failure modes such as overextending parameter ranges, dropping a crucial factor, altering quantifier order, and using the wrong asymptotic dependence on R.

A strong MCQ with no answer leakage and high-quality near-miss distractors. It avoids tautology reasonably well, though it tests careful theorem discrimination more than deep generative reasoning.