The quantic monoid and degenerate quantized enveloping algebras

Monoidal structure of the category of u$_q^+$-modules

We study the finite dimensional modules on the half-quantum group u_q^+ at a root of unity q, whose action can be extended to u_q (quotient of the quantized enveloping algebra of sl_2). We derive decomposition formulas of the tensor product of indecomposable u_q^+-modules, which includes the cases of the universal and the quantized universal enveloping algebra of sl_2 for q not a root of unity. We also prove that simple modules on u_q correspond exactly to the extendable non projective u_q^+-modules. We thus establish decomposition formulas for the tensor product of simple u_q-modules.

量子包络代数上的辫子交换代数

Braided commutative algebras over quantized enveloping algebras

本研究推广Davydov全中心构造，从B-中心幺半范畴的代数出发在相对幺半中心中构建辫子交换代数，将其应用于量子包络代数表示范畴得到辫子交换模代数，还构造了B-中心幺半范畴中代数的Morita不变量。

On the center of the quantized enveloping algebra of a simple Lie algebra

Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the quantum group $U_q(\frak{g})$ is isomorphic to a monoid algebra, and that $Z(U_q(\frak{g}))$ is a polynomial algebra if and only if $\frak{g}$ is of type $A_1, B_n, C_n, D_{2k+2}, E_7, E_8, F_4$ or $G_2.$ Moreover, in case $\frak{g}$ is of type $D_{n}$ with $n$ odd, then $Z(U_q(\frak{g}))$ is isomorphic to a quotient algebra of a polynomial algebra in $n+1$ vari…

Strongly homotopy Lie algebras

The present paper can be thought of as a continuation of the paper "Introduction to sh Lie algebras for physicists" by T. Lada and J. Stasheff (International Journal of Theoretical Physics Vol. 32, No. 7 (1993), 1087--1103, appeared also as preprint hep-th/9209099) which provided an exposition of the basic ingredients of the theory of strongly homotopy Lie algebras sufficient for the underpinnings of the physically relevant examples. We demonstrate the `strong homotopy' analog of the usual relation between Lie and associative algebras and investigate the universal enveloping algebra functor em…

Cohomological construction of quantized universal enveloping algebras

Given an associative algebra $A$, and the category, $\cC$, of its finite dimensional modules, additional structures on the algebra $A$ induce corresponding ones on the category $\cC$. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on $Rep_A$ is induced by an algebra homomorphism $A\to A\otimes A$ (comultiplication), coassociative up to conjugation by $Φ\in A^{\otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $\cR\in A^{\otimes 2}$ (commutativity constraint), together with an antiautomorphism (antipode), $S$, of $A$ satisfying the certain com…

Categorification of the adjoint action of quantum groups

Let $U$ be a quantized enveloping algebra. We consider the adjoint action of an $\mathfrak{sl}_2$-subalgebra of $U$ on a subalgebra of $U^+$ that is maximal integrable for this action. We categorify this representation in the context of quiver Hecke algebras. We obtain an action of the 2-category associated with $\mathfrak{sl}_2$ on a category of modules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of the main new features is a compatibility of the cat…