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Register a new userModule categories over affine supergroup schemes
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Authors
Shlomo Gelaki
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Let $k$ be an algebraically closed field of characteristic $0$ or $p>2$. Let $mathcal{G}$ be an affine supergroup scheme over $k$. We classify the indecomposable exact module categories over the tensor category ${rm sCoh}_{rm f}(mathcal{G})$ of (coherent sheaves of) finite dimensional $mathcal{O}(mathcal{G})$-supermodules in terms of $(mathcal{H},Psi)$-equivariant coherent sheaves on $mathcal{G}$. We deduce from it the classification of indecomposable {em geometrical} module categories over $sRep(mathcal{G})$. When $mathcal{G}$ is finite, this yields the classification of {em all} indecomposable exact module categories over the finite tensor category $sRep(mathcal{G})$. In particular, we obtain a classification of twists for the supergroup algebra $kmathcal{G}$ of a finite supergroup scheme $mathcal{G}$, and then combine it with cite[Corollary 4.1]{EG3} to classify finite dimensional triangular Hopf algebras with the Chevalley property over $k$.
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We study monoidal categorifications of certain monoidal subcategories $mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $mathcal{C}_{mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $mathcal{C}_{mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q(mathfrak{g})$-modules. For a suitable infinite sequence $widehat{w}_0= cdots s_{i_{-1}}s_{i_0}s_{i_1} cdots$ of simple reflections, we introduce subcategories $mathcal{C}_{mathfrak{g}}^{[a,b]}$ of $mathcal{C}_{mathfrak{g}}^0$ for all $a le b in mathbb{Z}sqcup{ pm infty }$. Associated with a certain chain $mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(mathfrak{C})$ in $mathcal{C}_{mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $mathcal{C}_{mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(mathcal{C}_{mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $mathcal{C}_{mathfrak{g}}^-$ by Hernandez-Leclerc since it is $mathcal{C}_{mathfrak{g}}^{[-infty,0]}$, and is also applicable to $mathcal{C}_{mathfrak{g}}^0$ since it is $mathcal{C}_{mathfrak{g}}^{[-infty,infty]}$.
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${mathcal C}$ to be equivalent. This concludes the classification of such module categories.
We generalize the definition of an exact sequence of tensor categories due to Brugui`eres and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle term of an exact sequence is semisimple if so are the other two terms.
Let $Vsubseteq A$ be a conformal inclusion of vertex operator algebras and let $mathcal{C}$ be a category of grading-restricted generalized $V$-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which $mathcal{C}$ inherits semisimplicity from the category of grading-restricted generalized $A$-modules in $mathcal{C}$, and vice versa. The most important condition is that $A$ be a rigid $V$-module in $mathcal{C}$ with non-zero categorical dimension, that is, we assume the index of $V$ as a subalgebra of $A$ is finite and non-zero. As a consequence, we show that if $A$ is strongly rational, then $V$ is also strongly rational under the following conditions: $A$ contains $V$ as a $V$-module direct summand, $V$ is $C_2$-cofinite with a rigid tensor category of modules, and $A$ has non-zero categorical dimension as a $V$-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that $A$ is a vertex operator superalgebra.
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