相关论文: An Infinite-Dimensional $\square_q$-Module Obtaine…
Let $\mathbb{F}$ denote an algebraically closed field with characteristic $0$, and let $q$ denote a nonzero scalar in $\mathbb{F}$ that is not a root of unity. Let $\mathbb{Z}_4$ denote the cyclic group of order $4$. Let $\square_q$ denote…
We consider the algebra $\square_q$ which is a mild generalization of the quantum algebra $U_q(\frak{sl}_2)$. The algebra $\square_q$ is defined by generators and relations. The generators are $\{x_i\}_{i\in \mathbb{Z}_4}$, where…
Let $K$ denote an algebraically closed field with characteristic 0, and let $q$ denote a nonzero scalar in $K$ that is not a root of unity. Let $A_q$ denote the unital associative $K$-algebra defined by generators $x,y$ and relations…
We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ with $q$ not a root of unity. We describe the finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules from the point of view of the equitable presentation.
We consider the quantized enveloping algebra $U_q(\widehat{\mathfrak{sl}}_2)$ and its basic module $V(\Lambda_0)$. This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe $V(\Lambda_0)$ using a…
Let $U_q(\hat{sl}_2)^{\geq 0}$ denote the Borel subalgebra of the quantum affine algebra $U_q(\hat{sl}_2)$. We show that the following hold for any choice of scalars $\epsilon_0, \epsilon_1$ from the set ${1,-1}$. (i) Let $V$ be a…
Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote a nonzero scalar in $\mathbb F$ that is not a root of unity. The universal DAHA (double affine Hecke algebra) $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is a unital…
We prove a bijection between finite-dimensional irreducible modules for an arbitrary quantum affine algebra $U_q(g)$ and finite-dimensional irreducible modules for its Borel subalgebra $U_q(g)^{\geq 0}$.
Let $\mathbb F$ denote an algebraically closed field and assume that $q\in \mathbb F$ is a primitive $d^{\rm \, th}$ root of unity with $d\not=1,2,4$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb…
We give a classification of all irreducible completely pointed $U_q(\mathfrak{sl}_{n+1})$ modules over a characteristic zero field in which $q$ is not a root of unity. This generalizes the classification result of Benkart, Britten and…
Since the introduction of Askey-Wilson algebras by Zhedanov in 1991, the classification of the finite-dimensional irreducible modules of Askey-Wilson algebras remains open. A universal analog $\triangle_q$ of the Askey-Wilson algebras was…
All finite dimensional Nichols algebras with diagonal type of connected finite dimensional Yetter-Drinfeld modules over finite cyclic group $\mathbb Z_n$ are found. It is proved that finite dimensional Nichols algebra over $\mathbb Z_2$ is…
The category of finite dimensional module over the quantum superalgebra U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even when q is not a…
The zero modes of the chiral SU(n) WZNW model give rise to an intertwining quantum matrix algebra A generated by an n x n matrix a=(a^i_\alpha) (with noncommuting entries) and by rational functions of n commuting elements q^{p_i}. We study…
Under the assumption that the quantum parameter $q$ is an $l$-th primitive root of unity with $l$ odd in a field $F$ of characteristic 0 and $m+n\geq r$, we obtained a complete classification of irreducible modules of the $q$-Schur…
We construct a class of modules for extended affine Lie algebra $\widetilde{\frak{gl}_l({\bc_q})}$ by using the free fields. A necessary and sufficient condition is given for those modules being irreducible.
Recently B. Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra via generators and relations. To obtain this presentation they defined an algebra $\boxtimes$ by generators and relations, and displayed…
The simple integrable modules with finite dimensional weight spaces are classified for the quantum affine special linear superalgebra $\U_q(\hat{\mathfrak{sl}}(M|N))$ at generic $q$. Any such module is shown to be a highest weight or lowest…
Suppose that $Q$ is a connected quiver without oriented cycles and $\sigma$ is an automorphism of $Q$. Let $k$ be an algebraically closed field whose characteristic does not divide the order of the cyclic group $\langle\sigma\rangle$. The…
The affine quantum Schur algebra is a certain important infinite dimensional algebra whose representation theory is closely related to that of quantum affine $\frak{gl}_n$. Finite dimensional irreducible modules for the affine quantum Schur…