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Register a new userDrinfeld-Hecke algebras over cocommutative algebras
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Authors
Apoorva Khare
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If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have families of deformations that have been studied widely in the literature; examples include symplectic reflection algebras and infinitesimal Hecke algebras. We introduce a family of deformations of these smash product algebras for general A, and characterize the PBW property. We then characterize the Jacobi identity for grouplike algebras (that include group rings and the nilCoxeter algebra), and precisely identify the PBW deformations in the example where A is the nilCoxeter algebra. We end with the more prominent case - where A is a Hopf algebra. We show the equivalence of sever
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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.
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.
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Let $left( H,Rright) $ be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field $k$. In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category ${} {}_{H}^{H}mathcal{YD}.$ Let $H_{R}$ be the Majids transmuted braided group of $left( H,Rright) ,$ we show that $H_{R}$ is cosemisimple. As a coalgebra, let $H_{R}=D_{1}opluscdotsoplus D_{r}$ be the sum of minimal $H$-adjoint-stable subcoalgebras. For each $i$ $left( 1leq ileq rright) $, we choose a minimal left coideal $W_{i}$ of $D_{i}$, and we can define the $R$-adjoint-stable algebra $N_{W_{i}}$ of $W_{i}$. Using Ostriks theorem on characterizing module categories over monoidal categories, we prove that $Vin{}_{H}^{H}mathcal{YD}$ is irreducible if and only if there exists an $i$ $left( 1leq ileq rright) $ and an irreducible right $N_{W_{i}}$-module $U_{i}$, such that $Vcong U_{i}otimes_{N_{W_{i}}}left( Hotimes W_{i}right) $. Our structure theorem generalizes the results of Dijkgraaf-Pasquier-Roche and Gould on Yetter-Drinfeld modules over finite group algebras. If $k$ is an algebraically closed field of characteristic, we stress that the $R$-adjoint-stable algebra $N_{W_{i}}$ is an algebra over which the dimension of each irreducible right module divides its dimension.
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