The Casimir elements of the Racah algebra

Let [Formula: see text] denote a field with [Formula: see text]. The Racah algebra [Formula: see text] is the unital associative [Formula: see text]-algebra defined by generators and relations in the following way. The generators are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The relations assert that [Formula: see text] and each of the elements [Formula: see text] is central in [Formula: see text]. Additionally, the element [Formula: see text] is central in [Formula: see text]. We call each element in [Formula: see text] a Casimir element of [Formula: see text], where [Formula: see text] is the commutative subalgebra of [Formula: see text] generated by [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The main results of this paper are as follows. Each of the following distinct elements is a Casimir element of [Formula: see text]: [Formula: see text] [Formula: see text] [Formula: see text] The set [Formula: see text] is invariant under a faithful [Formula: see text]-action on [Formula: see text]. Moreover, we show that any Casimir element [Formula: see text] is algebraically independent over [Formula: see text]; if [Formula: see text], then the center of [Formula: see text] is [Formula: see text].

New realizations of algebras of the Askey–Wilson type in terms of Lie and quantum algebras

The Askey–Wilson algebra is realized in terms of the elements of the quantum algebras [Formula: see text] or [Formula: see text]. A new realization of the Racah algebra in terms of the Lie algebras [Formula: see text] or [Formula: see text] is also given. Details for different specializations are provided. The advantage of these new realizations is that one generator of the Askey–Wilson (or Racah) algebra becomes diagonal in the usual representation of the quantum algebras whereas the second one is tridiagonal. This allows us to recover easily the recurrence relations of the associated orthogonal polynomials of the Askey scheme. These realizations involve rational functions of the Cartan generator of the quantum algebras, where they are linear with respect to the other generators and depend on the Casimir element of the quantum algebras.

The Cubic Dirac Operator for Infinite-Dimensonal Lie Algebras

Let be an infinite-dimensional graded Lie algebra, with dim , equipped with a non-degenerate symmetric bilinear form B of degree 0. The quantum Weil algebra is a completion of the tensor product of the enveloping and Clifford algebras of g. Provided that the Kac–Peterson class of g vanishes, one can construct a cubic Dirac operator D 2 , whose square is a quadratic Casimir element. We show that this condition holds for symmetrizable Kac– Moody algebras. Extending Kostant's arguments, one obtains generalized Weyl–Kac character formulas for suitable “equal rank” Lie subalgebras of Kac–Moody algebras. These extend the formulas of G. Landweber for affine Lie algebras.

Commuting Families in Hecke and Temperley-Lieb Algebras

We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

ISOTROPY REPRESENTATIONS, EIGENVALUES OF A CASIMIR ELEMENT, AND COMMUTATIVE LIE SUBALGEBRAS

Let [hfr ] be a reductive subalgebra of a semisimple Lie algebra [gfr ] and C[hfr ] ∈ U([hfr ]) be the Casimir element determined by the restriction of the Killing form on [gfr ] to [hfr ]. The paper studies eigenvalues of C[hfr ] on the isotropy representation [mfr ]≃[gfr ]/[hfr ]. Some general estimates connecting the eigenvalues and the Dynkin indices of [mfr ] are given. If [hfr ] is a symmetric subalgebra, it is shown that describing the maximal eigenvalue of C[hfr ] on exterior powers of [mfr ] is connected with possible dimensions of commutative Lie subalgebras in [mfr ], thereby extending a result of Kostant. In this situation, a formula is also given for the maximal eigenvalue of C[hfr ] on ∧ [mfr ]. More generally, a similar picture arises if [hfr ] = [gfr ]Θ, where Θ is an automorphism of finite order m and [mfr ] is replaced by the eigenspace of Θ corresponding to a primitive mth root of unity.

VOLKOV'S ALGEBRA AND CASIMIR ELEMENT OF SL2

We construct 3-D module of sl 2 and then obtain the generator of classical Volkov's algebra as Casimir element of sl 2. Only classical level is discussed.