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Families of operator identities appeared as a consequence of an existence of finite-dimensional representation of (super) Lie algebras of first-order differential operators and $q$-deformed (quantum) algebras of first-order…

The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also…

We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class…

We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…

Let $\Phi:V\to V\otimes U$ be an intertwining operator between representations of a simple Lie algebra (quantum group, affine Lie algebra). We define its generalized character to be the following function on the Cartan subalgebra with…

An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

We prove a factorization-concentration result for characters of symmetric groups. This is then applied to the asymptotic behaviour of the decomposition of the tensor representations. There are connections with the Pastur-Marcenko…

We obtain sharp uniform bounds on the low lying eigenfunctions for a class of semiclassical pseudodifferential operators with double characteristics and complex valued symbols, under the assumption that the quadratic approximations along…

We generalize several important results from the perturbation theory of linear operators to the setting of semisimple orthogonal symmetric Lie algebras. These Lie algebras provide a unifying framework for various notions of matrix…

From an analytical perspective, we introduce a sequence of Cartier operators that act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic $p$. In this work, we discover the binomial…

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in…

We extend to a scheme-theoretic context the notion of a combinatorial differential form, due to A.Kock in the framework of synthetic differential geometry. We show that group-valued combinatorial forms on a scheme may be identified, under…

Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation were found. In particular, the Lie algebra of the multiplication…

A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…

In recent years, the notion of characteristic polynomial of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the…

The main purpose of the present work is solving the characterization problem which consist of identification of necessary and sufficient conditions on the scattering data ensuring that the reconstructed potential belongs to a particular…

Foundational material on complex Lie supergroups and their radial operators is presented. In particular, Berezin's recursion formula for describing the radial parts of fundamental operators in general linear and ortho-symplectic cases is…

We show that two approaches to equivariant strict deformation quantization of C*-algebras by actions of negatively curved Kahlerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual…

This paper deals with well-known higher-order generalizations of Hankel operators. We show that higher-order Hankel operators can be written explicitly as linear differential operators, and give the exact form of these differential…

Many interesting and useful symbolic computation algorithms manipulate mathematical expressions in mathematically meaningful ways. Although these algorithms are commonplace in computer algebra systems, they can be surprisingly difficult to…