Related papers: On multigraded generalizations of Kirillov-Resheti…

We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.

We study locally finitary realizations of simple transitive module categories of infinite rank over the monoidal category $\mathscr{C}$ of finite dimensional modules for the complex Lie algebra $\mathfrak{sl}_2$. Combinatorics of such…

We prove that the tensor product of a simple and a finite dimensional $\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $\mathfrak{q}(n)$-supermodules to that of simple…

We construct all finite irreducible modules over Lie conformal superalgebras of type K

When a non-singular complex projective surface $X$ satisfies that $K_X\sim 0$, we shall show that there are only finitely many isomorphic classes as abstract schemes in the set of moduli scheme of $H$-semistable sheaves with fixed Chern…

We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras.…

Let $R$ be a commutative ring. A full additive subcategory $\C$ of $R$-modules is triangulated if whenever two terms of a short exact sequence belong to $\C$, then so does the third term. In this note we give a classification of…

We propose a very general construction of simple Virasoro modules generalizing and including both highest weight and Whittaker modules. This reduces the problem of classification of simple Virasoro modules which are locally finite over a…

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $\overline{U}_q^H(\mfg)$ of a simple Lie algebra $\mfg$ at roots of unity, and study their categories…

It is proved that uniformly bounded simple modules over higher rank super-Virasoro algebras are modules of the intermediate series, and that simple modules with finite dimensional weight spaces are either modules of the intermediate series…

We study modules over a generalized Weyl algebra $R(\sigma,a)$ which are free when restricted to the base ring $R$. When $R$ is an integral domain, we construct all such finite-rank modules up to isomorphism, leading to new simple modules…

Let $K$ be any field, and let $E$ be any graph. We explicitly construct the projective resolution of simple left modules over the Leavitt path algebra $L_K(E)$ associated to cycles and irreducible polynomials. Then we study the dimension of…

We show the denominator formulas for the normalized $R$-matrix involving two arbitrary Kirillov--Reshetikhin (KR) modules $W^{(k)}_{m,a}$ and $W^{(l)}_{p,b}$ in all nonexceptional affine types, $D_4^{(3)}$, and $G_2^{(1)}$. To achieve our…

We derive decomposition formulas for supercharacters of quantum affine ortho-symplectic superalgebras and twisted quantum affine superalgebras into supercharacters of their finite-type quantum sub-superalgebras, by employing Cauchy-type…

Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over…

Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the…

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them…

Let $\mathcal{O}_q(G)$ be the quantized algebra of regular functions on a semisimple simply connected compact Lie group $G$. Simple unitarizable left $\mathcal{O}_q(G)$-module are classified. In this article, we compute their…

The Weyl algebra over a field $k$ of characteristic $0$ is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all $\mathbb{Z}$-graded simple rings of GK-dimension 2 and show that they…

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be…