Related papers: Casimir operators for Lie superalgebras
A tensor extension of the Poincar\'e algebra is proposed for the arbitrary dimensions. Casimir operators of the extension are constructed. A possible supersymmetric generalization of this extension is also found in the dimensions $D=2,3,4$.
We prove the equivalence of two presentations of deformed double current algebras associated to a complex simple Lie algebra, the first one obtained via a degeneration of affine Yangians while the other one naturally appeared in the…
Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest weight representation.
Let $U_q(\hat{\cal G})$ be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation.…
We propose an algorithm for obtaining the spectra of Casimir (Laplace) operators on Lie groups. We prove that the existence of the normal polarization associated with a linear functional on the Lie algebra is necessary and sufficient for…
A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic $p.$ We generalize this result to a class of Lie algebras with a property that they arise as…
In recent years, algebras and modules of differential operators have been extensively studied. Equivariant quantization and dequantization establish a tight link between invariant operators connecting modules of differential operators on…
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…
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.
The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g=gl(n,2m),…
We investigate the nonlinear algebra $W_3$ generated by the 9 functionally independent permutation-symmetric operators in the three-particle rational quantum Calogero model. Decoupling the center of mass, we pass to a smaller algebra $W'_3$…
This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit…
Over an algebraically closed fields, an alternative to the method due to Kostrikin and Shafarevich was recently suggested. It produces all known simple finite dimensional Lie algebras in characteristic p>2. For p=2, we investigate one of…
We study the algebraic structure of the Poisson algebra P(O) of polynomials on a coadjoint orbit O of a semisimple Lie algebra. We prove that P(O) splits into a direct sum of its center and its derived ideal. We also show that P(O) is…
We describe the centers of the universal enveloping algebras of nilpotent Lie algebras of dimension at most six over fields of prime characteristic. If the characteristic is not smaller than the nilpontency class, then the center is the…
We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and its eigenvector $v$. This pair allows to build a Lie bracket on…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…
From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4,…
In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras…
We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type…