Related papers: Eigenvalues of Casimir operators for $gl(m/\infty)…
A full set of (higher order) Casimir invariants for the Lie algebra $gl(\infty )$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight (unitarizable) irreducible representations with only a…
For each quantum superalgebra $U_q[osp(m|n)]$ with $m>2$, an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary…
Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest weight representation.
We present the eigenvalues of the Casimir invariants for the type I quantum superalgebras on any irreducible highest weight module.
Maass forms for $SL(n,\mathbb{Z})$ are defined to be eigenfunctions of the Casimir operators $\mathcal{D}_{m,n}$ of orders $1 \leq m \leq n$ for $GL(n,\mathbb{R})$. For any $1 \leq m \leq n$ and Maass form $\phi$ for $SL(n,\mathbb{Z})$, we…
Let $\g$ be a finite-dimensional complex semisimple Lie algebra and $\b$ a Borel subalgebra. Then $\g$ acts on its exterior algebra $\w\g$ naturally. We prove that the maximal eigenvalue of the Casimir operator on $\w\g$ is one third of the…
It is given a way of computing Casimir eigenvalues for Weyl orbits as well as for irreducible representations of Lie algebras. A kappa(s) number of polinomials which depend on rank N are obtained explicitly for A_N Casimir operators of…
Two classes of irreducible highest weight modules of the general linear Lie superalgebra $gl(1/\infty)$ are constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the algebra…
For two different natural definitions of Casimir operators for simple Lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal Vogel's parameters $\alpha, \beta, \gamma$ and…
It is shown, for any irreducible representation of $E_8$ Lie algebra, that eigenvalues of Casimir operators can be calculated in the form of invariant polinomials which are decomposed in terms of $A_8$ basis functions. The general method is…
We give a general method to construct a complete set of linearly independent Casimir operators of a Lie algebra with rank N. For a Casimir operator of degree p, this will be provided by an explicit calculation of its symmetric coefficients…
The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is well-known now for powers up to the fourth. The paper describes an extension of this uniformity for the…
We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite…
In this paper fundamental Wigner coefficients are determined algebraically by considering the eigenvalues of certain generalized Casimir invariants. Here this method is applied in the context of both type 1 and type 2 unitary…
Let G be symmetrizable Kac-Moody Lie algebra. In this paper we describe a new class of central operators generalising the Casimir operator. We also prove some properties of these operators and show that these operators move highest weight…
In this paper, we study a certain specialization of Okounkov polynomials and its connection with the eigenfunctions of higher Casimir operators for the complex semi-simple lie algebra gl_n.
We construct explicit formulae for the eigenvalues of certain invariants of the Lie superalgebra gl(m|n) using characteristic identities. We discuss how such eigenvalues are related to reduced Wigner coefficients and the reduced matrix…
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…
A class of highest weight irreducible representations of the quantum algebra U_h(gl_\infty) is constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators…
This paper deals with the striking fact that there is an essentially canonical path from the $i$-th Lie algebra cohomology cocycle, $i=1,2,... l$, of a simple compact Lie algebra $\g$ of rank $l$ to the definition of its primitive Casimir…