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.Comment: 22 pages; v2: minor revisions as suggested by the refere

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.Comment: 27 page

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.Comment: 28 pages, Remarks 4.6 and 4.7 added at the suggestion of the refere

Morita equivalence for group -theoretical categories

We give necessary and sufficient conditions for two pointed categories to be dual to each other with respect to a module category. Whenever the dual of a pointed 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, o), where G is a finite group and o ∈ H3(G, kx). A group-theoretical and cohomological interpretation of this relation is given. As an application, we give a series of concrete examples of pairs of groups that are categorically Morita equivalent but have non-isomorphic Grothendieck rings. 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. The notion of a nilpotent fusion category, which categorically extends the notion of a nilpotent group, was introduced by Gelaki and Nikshych. We give sufficient conditions for a group-theoretical category to be nilpotent. We classify Lagrangian subcategories of the representation category of a twisted quantum double Do( G), where G is a finite group and o is a 3-cocycle on it. This gives a description of all braided tensor equivalences between twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of Rep(Do( G)) and module categories over the category VecwG of twisted G-graded vector spaces such that the dual fusion category is pointed. As a consequence, we establish that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories

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.Comment: 14 pages; v2: Sections 4 and 5 of v1 merged, minor revisions; to appear in Proc. Amer. Math. So

ECHOING QUALITIES OF REDUCED HINDRANCE AND ELEVATED VERACITY ON SAME CHANNEL

Two fundamental needs are low delay and data integrity. However, in many situations, both of these needs can't be satisfied concurrently. Within this paper, in line with the idea of potential in physics, we advise IDDR, a multi-path dynamic routing formula, to solve this conflict. Applications running on a single Wireless Sensor Network (WSN) platform will often have different Service quality (QoS) needs. By setting up a virtual hybrid potential field, IDDR separates packets of applications with various QoS needs based on the weight allotted to each packet, and routes them for the sink through different pathways to enhance the information fidelity for integrity-sensitive applications in addition to lessen the finish-to-finish delay for delay-sensitive ones. Simulation results show IDDR provides data integrity and delay differentiated services. While using Lyapunov drift technique, we prove that IDDR is stable

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.Comment: 26 pages; several comments and references adde