arXiv subjects

Deepak Naidu

Publications and source records attributed to Deepak Naidu.

11 recordsLinked to original sources

arXiv2014

Hochschild cohomology and quantum Drinfeld Hecke algebras

Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups and we exploit computations from our paper with Shroff for diagonal actions. By combining our work with recent results of Levandovskyy and Shepler, we produce examples of quantum Drinfeld Hecke algebras. These algebras generalize the braided Cherednik algebras of Bazlov and Berenstein.

math.RA↗

arXiv2013

Classification of integral modular categories of Frobenius-Perron dimension pq^4 and p^2q^2

We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.

math.QA↗

arXiv2012

Twisted quantum Drinfeld Hecke algebras

We generalize quantum Drinfeld Hecke algebras by incorporating a 2-cocycle on the associated finite group. We identify these algebras as specializations of deformations of twisted skew group algebras, giving an explicit connection to Hochschild cohomology. We classify these algebras for diagonal actions, as well as for the symmetric groups with their natural representations. Our results show that the parameter spaces for the symmetric groups in the twisted setting is smaller than in the untwisted setting.

math.RA↗

arXiv2011

Crossed pointed categories and their equivariantizations

We propose the notion of quasi-abelian third cohomology of crossed modules, generalizing Eilenberg and MacLane's abelian cohomology and Ospel's quasi-abelian cohomology, and classify crossed pointed categories in terms of it. We apply the process of equivariantization to the latter to obtain braided fusion categories which may be viewed as generalizations of the categories of modules over twisted Drinfeld doubles of finite groups. As a consequence, we obtain a description of all braided group-theoretical categories. A criterion for these categories to be modular is given. We also describe the quasi-triangular quasi-Hopf algebras underlying these categories.

math.QA↗

arXiv2010

Hochschild cohomology of group extensions of quantum symmetric algebras

Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.

math.RA↗

arXiv2009

A finiteness property for braided fusion categories

We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron dimension are precisely those with property F.

math.QA↗

arXiv2009

Centers of graded fusion categories

Let C be a fusion category faithfully graded by a finite group G and let D be the trivial component of this grading. The center Z(C) of C is shown to be canonically equivalent to a G-equivariantization of the relative center Z_D(C). We use this result to obtain a criterion for C to be group-theoretical and apply it to Tambara-Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara-Yamagami categories. Finally, we prove a general result about existence of zeroes in S-matrices of weakly integral modular categories.

math.QA↗

arXiv2009

Fusion subcategories of representation categories of twisted quantum doubles of finite groups

We describe all fusion subcategories of the representation category of a twisted quantum double of a finite group. In view of the fact that every group-theoretical braided fusion category can be embedded into a representation category of a twisted quantum double of a finite group, this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of representation category of a twisted quantum double of a finite group. We also give a characterization of group-theoretical braided fusion categories as equivariantizations of pointed categories.

math.QA↗

arXiv2007

Some properties of group-theoretical categories

We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit description of the simple objects in a group-theoretical category (following Ostrik, arXiv:math/0202130) and of the group of invertible objects of a group-theoretical category, in group-theoretical terms. Finally, under certain restrictive conditions, we describe the universal grading group of a group-theoretical category.

math.QA↗

arXiv2007

Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups

We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.

math.QA↗

arXiv2006

Categorical Morita equivalence for group-theoretical categories

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, w), where G is a finite group and w is a 3-cocycle on G. A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have non-isomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data.

math.QA↗