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We investigate the rate of growth of the function of n which counts the number of complex irreducible representations of a fixed group of degree less than or equal to n. The emphasis is on linear groups, especially compact real and p-adic…
We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of $Q \in \C$, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models…
Orthosymplectic Lie superalgebras are fundamental symmetries in modern physics, such as massive supergravity. However, their representations are far from being thoroughly understood. In the present paper, we completely determine the…
We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p>2. A superalgebra generalization of the celebrated Kac-Weisfeiler Conjecture is formulated, which exhibits a…
We study the interplay between double cross sum decompositions of a given Lie algebra and classical r-matrices for its semidual. For a class of Lie algebras which can be obtained by a process of generalised complexification we derive an…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
We classify irreducible finite-dimensional modules of a collection of real Lie superalgebras that includes the simple ones, their classical variants, complex Lie superalgebras after restriction of scalars, and all real Lie algebras. Our…
We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or…
The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on…
The motivation for this paper is to detect when an irreducible projective variety V is not toric. We do this by analyzing a Lie group and a Lie algebra associated to V. If the dimension of V is strictly less than the dimension of the above…
We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such…
Recently one of the authors obtained a classification of simple infinite-dimensional Lie superalgebras of vector fields which extends the well-known classification of E. Cartan in the Lie algebra case. The list consists of many series…
We present some recently discovered infinite dimensional Lie algebras that can be understood as extensions of the algebra Map(M,g) of maps from a compact p-dimensional manifold to some finite dimensional Lie algebra g. In the first part of…
We carry out the complete group classification of the class of (1+1)-dimensional linear Schr\"odinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we…
We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is…
The article is devoted to some ``strange'' phenomena of representation theory and their interrelations. Cross-projective representations of pairs of anticommutative algebras, alloys, their universal envelopping Lie algebras and their…
The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e. filiform Lie (super)algebras, into the theory of Lie algebras of order F$. Thus, the concept of filiform Lie algebras of order F is…
We continue our study of Hilbert space representations of the Reflection Equation Algebra, again focusing on the algebra constructed from the $R$-matrix associated to the $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We develop a form…
Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy…
In this paper we study the representation theory for certain ``half lattice vertex algebras.'' In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these…