Related papers: Finite type modules and Bethe Ansatz for quantum t…
In this paper, we first construct a Lie algebra $L$ from rank 3 quantum torus, and show that it is isomorphic to the core of EALAs of type $A_1$ with coordinates in rank 2 quantum torus. Then we construct two classes of irreducible ${\bf…
The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a…
We show that the Weyl-Kac type character formula holds for the integrable highest weight modules over the quantized enveloping algebra of any symmetrizable Kac-Moody Lie algebra, when the parameter $q$ is not a root of unity.
We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra $gl_N$. The models are defined on a tensor product irreducible $gl_N$-modules. For each model, there exist $N$ one-parameter families of commuting…
We study a relationship between the graded characters of generalized Weyl modules $W_{w \lambda}$, $w \in W$, over the positive part of the affine Lie algebra and those of specific quotients $V_{w}^- (\lambda) / X_{w}^- (\lambda)$, $w \in…
Let $V(\Lambda_i)$ (resp., $V(-\Lambda_j)$) be a fundamental integrable highest (resp., lowest) weight module of $U_q(\hat{sl}_{2})$. The tensor product $V(\Lambda_i)\otimes V(-\Lambda_j)$ is filtered by submodules…
We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics…
We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree.…
Let $\mathfrak g$ be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra {\bf W}$(n)$. We study weight $\mathfrak g$-modules using a method inspired by Mathieu's classification of the simple weight modules with finite…
By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over the algebraic closure ${{\bar \BF}}_q$ of the finite field $\BF_q$ of $q$ elements, we establish a relation between the representation theory of $A$ over ${{\bar…
We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the…
The representations of a quiver Q over a field k have been studied for a long time. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when…
The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same $R$-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe…
We study a family of power series characterized by a system of recursion relations (Q-system) with a certain convergence property. We show that the coefficients of the series are expressed by the numbers which formally count the…
In this paper we seek geometric and invariant-theoretic characterizations of (Schur-)representation finite algebras. To this end, we introduce two classes of finite-dimensional algebras: those with the dense-orbit property and those with…
We establish the Bethe equation of the $\tau^{(2)}$-model in the $N$-state chiral Potts model (including the degenerate selfdual cases) with alternating vertical rapidities. The eigenvalues of a finite-size transfer matrix of the chiral…
In this thesis we classify modules over a Witt-type Lie algebra and superalgebra such that when considered as modules of $\mathcal{U}(\mathfrak{h})$ they are free of rank 1. We provide sufficient conditions for simplicity, and compute the…
In this paper, we classify the irreducible integrable modules with finite dimensional weight spaces and non-trivial $\widetilde{\mathfrak g}_c$-action for the nullity $2$ toroidal extended affine Lie algebra $\widetilde{\mathfrak g}$, where…
As every simple module of a quiver Hecke algebra appears as the image of the R-matrix defined on the convolution product of certain cuspidal modules, knowing the $\mathbb{Z}$-invariants of the R-matrices between cuspidal modules is quite…
We study the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules where $kG$ is the group algebra of a finite group $G$ over a field of characteristic $p>0$. When the representation type of the group algebra is not wild, the…