Related papers: Dual pairs in complex classical groups and Lie alg…

We give a classification of the principal and distinguished nilpotent pairs in all classical Lie algebras. As a classification of the principal pairs in the exceptional simple Lie algebras was obtained earlier (see Appendix to Ginzburg's…

For a semisimple Lie group $G$, we study Discrete Series representations with admissible branching to a symmetric subgroup $H$. This is done using a canonical associated symmetric subgroup $H_0$, forming a pseudo-dual pair with $H$, and a…

Let $g$ be a classical Lie algebra, i.e., either $gl_n$, $sp_n$, or $so_n$ and let $e\in g$ be a nilpotent element. We study various properties of centralisers $g_e$. The first four sections deal with rather elementary questions, like the…

Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a basic classical Lie superalgebra over an algebraically closed field \mathbb{K} whose characteristic p>0 is a good prime for \mathfrak{g}. Let G_{\bar{0}} be the…

We discuss some aspects and examples of applications of dual algebraic pairs $({\cal G}_1,{\cal G}_2)$ in quantum many-body physics. They arise in models whose Hamiltonians $H$ have invariance groups $G_i$. Then one can take ${\cal G}_1 =…

We study cohomology for classical Lie superalgebras $\mathfrak{g}$ (e.g. gl(m|n)) over the complex numbers. Using results from invariant theory, we show that there exist subsuperalgebras which detect the cohomology of $\mathfrak{g}.$…

The Howe duality between quantum general linear supergroups was firstly established by Y. Zhang via quantum coordinate superalgebras. In this paper, we provide two other approaches to this Howe duality. One is constructed by quantum…

Discrete quantum groups were introduced as duals of compact quantum groups by Podle\'s and Woronowicz in 1990. Shortly after, they were defined and studied intrinsically by Effros and Ruan, and by this author. In 1998, with the introduction…

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…

We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp.…

In this article, we define the capable pairs of Lie superalgebras. We classify all capable pairs of abelian and Heisenberg Lie superalgebras. After that we discuss on pairs of Lie superalgebras with derived subalgebra of dimension one and a…

Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p > 0. An interesting class of representations of G consists of those G-modules having a good filtration -- i.e. a filtration whose…

A general theorem due to Howe of dual action of a classical group and a certain non-associative algebra on a space of symmetric or alternating tensors is reformulated in a setting of second quantization, and familiar examples in atomic and…

We establish a new Howe duality between a pair of quantum queer superalgebras $(\mathrm{U}_{q^{-1}}(\mathfrak{q}_n), \mathrm{U}_q(\mathfrak{q}_m))$. The key ingredient is the construction of a non-commutative analogue…

In this article we prove a Howe correspondence for a family of representations of sl(2n), which was introduced by Benkart, Britten, and Lemire.

Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing $0$. A selection of examples is given. These Lie pairs comprise two…

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…

We develop an operator commutant version of the First Fundamental Theorem of invariant theory for the general linear quantum group $U_q(\mathfrak{gl}_n)$ by using a double centralizer property inside a quantized Clifford algebra. In…

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho}$ for each irreducible…

Let $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ be a $\mathbb Z_2$-grading of a classical Lie algebra such that $(\mathfrak{g}, \mathfrak{g}_0)$ is a classical symmetric pair. Let $G$ be a classical group with Lie algebra…