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Register a new userGreen rings of Drinfeld Doubles of Taft algebras
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In this article, we investigate the representation ring (or Green ring) of the Drinfeld double $D(H_n(q))$ of the Taft algebra $H_n(q)$, where $n$ is an integer with $n>2$ and $q$ is a root of unity of order $n$. It is shown that the Green ring $r(D(H_n(q)))$ is a commutative ring generated by infinitely many elements subject to certain relations.
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In this paper, we study the tensor structure of category of finite dimensional representations of Drinfeld quantum doubles $D(H_n(q))$ of Taft Hopf algebras $H_n(q)$. Tensor product decomposition rules for all finite dimensional indecomposable modules are explicitly given.
For the Drinfeld double $D_n$ of the Taft algebra $A_n$ defined over an algebraically closed field $mathbb k$ of characteristic zero using a primitive $n$th root of unity $q in mathbb k$ for $n$ odd, $nge3$, we determine the ribbon element of $D_n$ explicitly. We use the R-matrix and ribbon element of $D_n$ to construct an action of the Temperley-Lieb algebra $mathsf{TL}_k(xi)$ with $xi = -(q^{frac{1}{2}}+q^{-frac{1}{2}})$ on the $k$-fold tensor product $V^{otimes k}$ of any two-dimensional simple $D_n$-module $V$. When $V$ is the unique self-dual two-dimensional simple module, we develop a diagrammatic algorithm for computing the $mathsf{TL}_k(xi)$-action. We show that this action on $V^{otimes k}$ is faithful for arbitrary $k ge 1$ and that $mathsf{TL}_k(xi)$ is isomorphic to the centralizer algebra $text{End}_{D_n}(V^{otimes k})$ for $1 le kle 2n-2$.
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
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.
In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfeld double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfeld doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite arithmetic root system. Key words: Hopf algebra, quantum group, Weyl groupoid
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