Related papers: An Explicit Construction of Casimir Operators and …

In this paper, we study Lie superalgebras of $2\times 2$ matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional…

A covariant pseudodifferential calculus on Riemann surfaces, based on the Krichever-Novikov global picture, is presented. It allows defining scalar and matrix KP operators, together with their reductions, in higher genus. Globally defined…

We consider algebras acting on Schur and Q-Schur polynomials, corresponding to Kadomtsev-Petviashvili (KP) and BKP hierarchies. We present them in the spirit of affine Yangians, paying special attention to commutative subalgebras, box…

Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for…

We calculate the degree 2 and 6 Casimirs operators in explicit form, with the generators of G2 written in terms of the subalgebra A2

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…

We outline an abstract approach to the pseudo-differential Weyl calculus for operators in function spaces in infinitely many variables. Our earlier approach to the Weyl calculus for Lie group representations is extended to the case of…

We show that the In\"on\"u-Wigner contraction naturally associated to a reduction chain $\frak{s}\supset \frak{s}^{\prime}$ of semisimple Lie algebras induces a decomposition of the Casimir operators into homogeneous polynomials, the terms…

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3, 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e.,…

Poincare Polynomial of a Kac-Moody Lie algebra can be obtained by classifying the Weyl orbit $W(\rho)$ of its Weyl vector $\rho$. A remarkable fact for Affine Lie algebras is that the number of elements of $W(\rho)$ is finite at each and…

This survey paper is an exposition of old and recent results of Kostant and al. on the relationships between the exterior algebra of a simple Lie algebra and the action of the Casimir operator on it. Our exposition relies on u-cohomology…

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the…

Linear operators preserving the direct sum of polynomial rings P(m)\oplus P(n) are constructed. In the case |m-n|=1 they correspond to atypical representations of the superalgebra osp(2,2). For |m-n|=2 the generic, finite dimensional…

We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras,…

The infinitesimal generator of a one-parameter subgroup of the infinite dimensional rotation group associated with the complex Gelfand triple $ (E) \subset L^2(E^*, \mu) \subset (E)^* $ is of the form $$ R_\kappa = \int_{T\times T}…

This paper introduces and studies a class of Weyl-type algebras \(A_{p,t,\cA} = \Weyl{e^{\pm x^{p} e^{t x}},\; e^{\cA x},\; x^{\cA}}\) constructed over exponential-polynomial rings, where \(\FF\) is a field of characteristic zero, \(\cA\)…

All finite dimensional irreducible representations of the simple Lie-Kac super algebra SU(2/1) are explicitly constructed in the Chevalley basis as complex matrices. For typical representations, the distinguished Dynkin label is not…

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae…

Contractions of Lie algebras are combined with the classical matrix method of Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie…

A complete choice of generators of the center of the enveloping algebras of real quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known…