A Category of Wide Subcategories

Covering theory for linear categories with application to derived categories

In this paper, we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are nonzero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.

We study three related homological properties of modules in the BGG category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and complexity. We demonstrate connections between projective dimension and singularity of modules and blocks. Similarly we investigate the connection between complexity and atypicality. This creates concrete tools to describe singularity and atypicality as homological, and hence categorical, properties of a block. However, we also demonstrate how two integral blocks in category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent. This principle implies that category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l left-parenthesis m vertical-bar n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {gl}(m|n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can contain infinitely many non-equivalent blocks, which we work out explicitly for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l left-parenthesis 3 vertical-bar 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {gl}(3|1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . All of this is in sharp contrast with category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for Lie algebras, but also with the category of finite dimensional modules for superalgebras. Furthermore we characterise modules with finite projective dimension to be those with trivial associated variety. We also study the associated variety of Verma modules. To do this, we also classify the orbits in the cone of self-commuting odd elements under the action of an even Borel subgroup.

Exceptional sequences and Drinfeld double Hall algebras

The category of finite dimensional modules over the quantum superalgebra [Formula: see text] is not semi-simple and the quantum dimension of a generic [Formula: see text]-module vanishes. This vanishing happens for any value of [Formula: see text] (even when [Formula: see text] is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize [Formula: see text] to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of [Formula: see text] by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize [Formula: see text] to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative [Formula: see text]-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category 𝒪 associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category 𝒪 and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category 𝒪 for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra 𝔮(2).

The $q$-deformed associative algebra $U_q'(\mathfrak{so}_n)$ is an example of an $\imath$quantum group. Finite-dimensional simple modules over $U_q'(\mathfrak{so}_n)$ were constructed by Gavrilik and Klimyk which limit to finite-dimensional simple modules over the universal enveloping algebra $U(\mathfrak{so}_n)$ as $q \to 1$. These modules are examples of Gelfand-Tsetlin modules since they are equal to the direct sum of common finite-dimensional eigenspaces under the action of a commutative subalgebra called the Gelfand-Tsetlin subalgebra $\Gamma \subset U_q'(\mathfrak{so}_n)$. In this thesis we begin to study the category of Gelfand-Tsetlin modules over $U_q'(\mathfrak{so}_n)$, which contains the category of finite-dimensional modules. Following the example of Drozd, Futorny and Ovsienko for $U(\mathfrak{gl}_n)$; Mazorchuk and Turowska for $U_q(\mathfrak{gl}_n)$; and Mazorchuk for $U(\mathfrak{so}_n)$; we construct infinite-dimensional analogs to the Gavrilik-Klimyk classical $U_q'(\mathfrak{so}_n)$-modules by removing the conditions on the combinatorial patterns parameterizing the bases. We call these generic Gelfand-Tsetlin modules. We find upper bounds for lengths of these modules for $n=3$ and $n=4$. Taking the limit as $q \to 1$, we obtain generic Gelfand-Tsetlin modules over $U(\mathfrak{so}_n)$ whose matrix coefficients are rational functions, thus extending the work of Mazorchuk. We use these large new families of modules to embed $U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$ into skew group algebras of shift operators. These embeddings map the Gelfand-Tsetlin subalgebras to certain invariant algebras under the actions of the direct products of ($q$-)Weyl groups. Lastly, we use this fact to show the Gelfand-Tsetlin subalgebras are Harish-Chandra subalgebras, as defined by Drozd-Futorny-Ovsienko.

Whittaker coinvariants for GL(m|n)

It is usually not straightforward to work with the category of perverse sheaves on a variety using only its definition as a heart of a $t$-structure. In this paper, the category of perverse sheaves on a smooth toric variety with its orbit stratification is described explicitly as a category of finite-dimensional modules over an algebra. An analogous result is also established for various categories of equivariant perverse sheaves, which in particular gives a description of perverse sheaves on toric orbifolds, and we also compare the derived category of the category of perverse sheaves to the derived category of constructible sheaves.

Finite dimensional simple modules of deformed current Lie algebras

Analytical Mechanics. Foundations of Engineering Mechanics Series

LetXbe a topologically stratified space,pbe any perversity onXandkbe a field. We show that the category ofp-perverse sheaves onX, constructible with respect to the stratification and with coefficients ink, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only ifXhas finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

For ℳ and mathcal{N} finite module categories over a finite tensor category mathcal{C} , the category mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) of right exact module functors is a finite module category over the Drinfeld center mathcal{Z} ( mathcal{C} ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map mathcal{C} - mathcal{C} -bimodule functors to objects of mathcal{Z} ( mathcal{C} ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and mathcal{N} are exact mathcal{C} -modules and mathcal{C} is pivotal, then the mathcal{Z} ( mathcal{C} )-module mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) is exact. We compute its relative Serre functor and show that if ℳ and mathcal{N} are even pivotal module categories, then mathrm{mathcal{R}}{ex}_{mathcal{C}} (ℳ, mathcal{N} ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in mathcal{Z} ( mathcal{C} ).

Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒪</mml:mi></mml:math> for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math>. We announce a generalization of this result to the extension of the Hecke algebra associated to the normalizer of a reflection subgroup.

The category of weight modules of the simple affine vertex algebra of at an admissible level is neither finite nor semisimple and modules are usually not lower‐bounded and have infinite‐dimensional conformal weight subspaces. However, this vertex algebra enjoys a duality with , the simple principal ‐algebra of at level (the super conformal algebra) where the levels are related via . Every weight module of is lower‐bounded and has finite‐dimensional conformal weight spaces. The main technical result is that every weight module of is ‐cofinite. The existence of a vertex tensor category follows and the theory of vertex superalgebra extensions implies the existence of vertex tensor category structure on for any admissible level . As applications, the fusion rules of ordinary modules with any weight module are computed, and it is shown that is a ribbon category if and only if is, in particular, it follows that for admissible levels and and , the category is a ribbon category.