Related papers: Deformations of unitary Howe dual pairs

We study the Howe correspondence for unipotent representations of irreducible dual pairs $(G',G)=(\text{U}_m(\mathbb{F}_q),\text{U}_n(\mathbb{F}_q))$ and $(G',G)=(\text{Sp}_{2m}(\mathbb{F}_q),\text{O}^\epsilon_{2n}(\mathbb{F}_q))$, where…

We study the deformations of twisted harmonic maps $f$ with respect to the representation $\rho$. After constructing a continuous "universal" twisted harmonic map, we give a construction of every first order deformation of $f$ in terms of…

The current article continues our project on representation theory, Euler elements, causal homogeneous spaces and Algebraic Quantum Field Theory (AQFT). We call a pair (h,k) of Euler elements orthogonal if $e^{\pi i \ad h} k = -k$. We show…

To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_k, of the smash product of G with the polynomial algebra on V. The algebra H_k, called a symplectic reflection algebra,…

Let p be a prime, M be a finite group, F be the field with p elements, and V be an absolutely irreducible FM-module. Then V has a universal deformation ring R(M,V) whose structure is closely related to the first and second cohomology groups…

We initiate the study of deformations of groups in three-dimensional complex hyperbolic geometry. Let $$G=\left\langle \iota_1, \iota_2, \iota_3, \iota_4 \Bigg| \begin{array}{c} \iota_1^2= \iota_2^2 = \iota_3^2=\iota_4^2=id,\\ (\iota_1…

We establish a quantum analogue of the classical metaplectic Howe duality involving the pair of Lie algebras $(\mathfrak{sp}_{2n},\mathfrak{sl}_2)$ in the case when $n=1$. Our results yield commuting representations of the pair of…

Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra $\mathfrak{g}$. An important branching problem is to determine the finite-dimensional highest-weight…

Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J_0 and generalizing the Witten algebra, were introduced some time ago by Delbecq and Quesne. In the present paper, the problem of endowing some…

We study the algebraic structure and representation theory of the Hopf algebras ${}_J\mathcal{O}(G)_J$ when $G$ is an affine algebraic unipotent group over $\mathbb{C}$ with $\mathrm{dim}(G) = n$ and $J$ is a Hopf $2$-cocycle for $G$. The…

The symmetric group S_n possesses a nontrivial central extension, whose irreducible representations, different from the irreducible representations of S_n itself, coincide with the irreducible representations of a certain algebra A_n.…

It is shown that the deformed Heisenberg algebra involving the reflection operator R (R-deformed Heisenberg algebra) has finite-dimensional representations which are equivalent to representations of paragrassmann algebra with a special…

The SU$(1,1)$ group plays a fundamental role in various areas of physics, including quantum mechanics, quantum optics, and representation theory. In this work we revisit the holomorphic discrete series representations of SU$(1,1)$, with a…

Let $k$ be a field of characteristic $p>0$, and let $W$ be a complete discrete valuation ring of characteristic $0$ that has $k$ as its residue field. Suppose $G$ is a finite group and $G^{\mathrm{ab},p}$ is its maximal abelian $p$-quotient…

In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that…

We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on…

In this paper, we study the behavior of categorical actions of a Lie algebra $\mathfrak{g}$ under the deformation of their spectra. We give conditions under which the general point of a family of categorical actions of $\mathfrak{g}$ carry…

We consider the target space theory of bosonic and heterotic string theory to first order in $\alpha'$ compactified to three dimensions, using a formulation that is manifestly T-duality invariant under ${\rm O}(d,d,\mathbb{R})$ with $d=23$…

We study double affine Hecke algebras at roots of unity and their relations with deformed Hilbert schemes.

We apply methods from strict quantization of solvable symmetric spaces to obtain universal deformation formulae for actions of a class of solvable Lie groups. We also study compatible co-products by generalizing the notion of smash product…