Related papers: Dual pairs in $PGL(n,\mathbb{C})$

Let $\mathfrak{g}$ be a vector space and $[,],[,]'$ be a pair of Lie brackets on $\mathfrak{g}$. By definition they are compatible if $[,]+[,]'$ is again a Lie bracket. Such pairs play important role in bihamiltonian and $r$-matrix…

Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $p\geq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and…

The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does…

We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a…

We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. For this, we develop and…

All two-dimensional reproducing formulae, i.e. of $L^2({\mathbb R}^2)$, resulting out of restrictions of the projective metaplectic representation to connected Lie subgroups of $Sp(2,{\mathbb R})$ and of type $\mathcal{E}_2$, were listed…

Discrete quantum groups were introduced as duals of compact quantum groups by Podle\'s and Woronowicz in 1990. They have been studied intrinsically by Effros and Ruan (1994) and by the author (1996). In a more recent note (2025), we have…

This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $\mathbb P^1_C\smallsetminus\{0,\infty\}$, where $C$ is…

For the Lie algebra $\g$ of a connected infinite-dimensional Lie group~$G$, there is a natural duality between so-called semi-equicontinuous weak-*-closed convex Ad^*(G)-invariant subsets of the dual space $\g'$ and Ad(G)-invariant lower…

Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth…

For the classical compact Lie groups K = U(N) the autocorrelation functions of ratios of random characteristic polynomials are studied. Basic to our treatment is a property shared by the spinor representation of the spin group with the…

Let $p$ be an odd prime and let $G$ be a non-abelian finite $p$-group of exponent $p^2$ with three distinct characteristic subgroups, namely $1$, $G^p$, and $G$. The quotient group $G/G^p$ gives rise to an anti-commutative ${\mathbb…

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf \cst-algebras, and a…

Fix an integral semisimple element $\lambda$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $\lambda$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of…

Lie algebra contractions on the classical Drinfel'd Double of a given Lie bialgebra are introduced and compared to the usual Lie bialgebra contraction theory. The connection between both approaches turns out to be intimately linked to…

We classify quantum analogues of actions of finite subgroups G of SL_2(k) on commutative polynomial rings k[u,v]. More precisely, we produce a classification of pairs (H,R), where H is a finite dimensional Hopf algebra that acts inner…

The Drinfeld double structure underlying the Cartan series An, Bn, Cn, Dn of simple Lie algebras is discussed. This structure is determined by two disjoint solvable subalgebras matched by a pairing. For the two nilpotent positive and…

Several general properties, concerning reduction algebras - rings of definition and algorithmic efficiency of the set of ordering relations - are discussed. For the reduction algebras, related to the diagonal embedding of the Lie algebra…

A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup…

In this short note we expand on recent results on the degenerate principle series $I(s,\chi)$ of classical groups associated to $s\in \mathbb{C}$ and a quadratic character $\chi$. In particular, we strengthen the result for $s\in…