Related papers: Shapovalov elements of classical and quantum group…

A complex vector space $V$ is an \'etale $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit and $\dim G=\dim V$. Such a module is called super-\'etale if the stabilizer of a point in the open orbit is trivial. Popov proved…

In this paper we classify all simple weight modules for a quantum group $U_q$ at a complex root of unity $q$ when the Lie algebra is not of type $G_2$. By a weight module we mean a finitely generated $U_q$-module which has finite…

This paper studies rational functions $\mathfrak{J}_\alpha(q)$, which depend on a positive element $\alpha$ of the root lattice of a root system. These functions arise as Shapovalov pairings of Whittaker vectors in Verma modules of highest…

We reconstruct finite-dimensional quantum theory from categorical principles. That is, we provide properties ensuring that a given physical theory described by a dagger compact category in which one may `discard' objects is equivalent to a…

Let $W$ be the Weyl group corresponding to a finite dimensional simple Lie algebra $\mathfrak{g}$ of rank $\ell$ and let $m>1$ be an integer. In [I21], by applying cluster mutations, a $W$-action on $\mathcal{Y}_m$ was constructed. Here…

We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a…

We define a family of universal finite-dimensional highest weight modules for affine Lie algebras, we call these Weyl modules. We conjecture that these are the classical limits of the irreducible finite--dimensional representations of the…

Let $\{q_n^{(\alpha,\beta,m)}(x)\}_{n\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \begin{equation*} \langle f,g\rangle_{\alpha,\beta,m}=\sum_{k=0}^m \int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\,…

For a simple Lie algebra $\mathfrak{g}$ of type $A_n,B_n,C_n$ or $D_n$, we give a characterization of the set of dominant integral weights $\lambda$ such that for any rational point $\mu$ in the fundamental Weyl chamber, $2\lambda-\mu$ is a…

Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal (Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of random Hermitian matrices, are considered. Averages are shown to satisfy duality relations like…

We define a natural quantum analogue for the ${\cal Z}$ algebra, and which we refer to as the ${\cal Z}_q$ algebra, by modding out the Heisenberg algebra from the quantum affine algebra $U_q(\hat{sl(2)})$ with level $k$. We discuss the…

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an…

We develop a theory of weights for a quantum analogue of the symmetric pair (gl4,gl2 x gl2) realised as a quantum symmetric pair subalgebra. Based on Letzter's triangular decomposition we define Verma modules. Using magical operators that…

Recently obtained recurrence formulae for relativistic hydrogenic radial matrix elements are cast in a simpler and perhaps more useful form. This is achieved with the help of a new relation between the $r^a$ and the $\beta r^b$ terms…

Let $\mathfrak{g}$ be a simple complex Lie algebra of a classical type and $U_q(\mathfrak{g})$ the corresponding Drinfeld-Jimbo quantum group at $q$ not a root of unity. With every point $t$ of the fixed maximal torus $T$ of an algebraic…

We study the computation of canonical bases of sets of univariate relations $(p_1,\ldots,p_m) \in \mathbb{K}[x]^{m}$ such that $p_1 f_1 + \cdots + p_m f_m = 0$; here, the input elements $f_1,\ldots,f_m$ are from a quotient…

For a commutative finite $\mathbb{Z}$-algebra, i.e., for a commutative ring $R$ whose additive group is finitely generated, it is known that the group of units of $R$ is finitely generated, as well. Our main results are algorithms to…

We consider irreducible lowest-weight representations of Cherednik algebras associated to certain classes of complex reflection groups in characteristic p. In particular, we study maximal graded submodules of Verma modules associated to…

Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical…

This paper deals with the set of $\alpha\in{\mathbb{R}}$ such that $\alpha \zeta^{n} \bmod 1$ tends to $0$ for a fixed $\zeta\in{\mathbb{R}}$, which we call $\mathscr{M}_{\zeta}$. Predominately the case of Pisot numbers $\zeta$ is studied.…