We describe a generalization of Drinfeld’s description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald’s difference operators. 1. The center of Uq(g), where g is a simple Lie algebra. 1.1. Let g be a simple Lie algebra over C of rank r. Let h be a Cartan subalgebra in g. Let W be the Weyl...

We consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form F = Tr(Φ 1 (z 1). .. Φ n (z n)q −∂ e h) where Φ i are intertwiners between Verma modules and evaluation modules over an affine Lie algebrâ g, ∂ is the grading operator in a Verma module and h is in the Cartan ...

In [8] and [9], Voiculescu introduced free entropy for n–tuples of self–adjoint elements in a II1–factor, and used it to prove that free group factors, L(Fn), lack Cartan subalgebras [9]. In [4], S. Popa introduced a property for II1–factors, called property C. (See [5] for a paper related to [4].) Like Property Γ of Murray and von Neumann, this is an asymptotic commutivity property, but it is ...

1. Introduction. 1. Let g be a complex semi-simple Lie algebra and f) a Cartan subalgebra of g. Let ir\ be an irreducible representation of g, with highest weight X, on a finite dimensional vector space V\. A well known theorem of E. Cartan asserts that the highest weight, X, of w\ occurs with multi-plicity one. It has been a question of long standing to determine, more generally , the multipli...

A low energy bound in a class of chiral solitonic field theories related the infrared physics of the SU(N) Yang-Mills theory is established. 1. The model. Consider N−1 smooth fields na = na(x) in spacetime taking their values in the Lie algebra of SU(N). The fields are chosen to be commutative [na, nb] = 0 and orthonormal (na, nb) = δab with respect the Cartan-Killing form in the Lie algebra. F...

Based on the theory of intuitionistic fuzzy sets, the concepts of intuitionistic fuzzy subalgebras with thresholds (λ, μ) and intuitionistic fuzzy ideals with thresholds (λ, μ) of BCI-algebras are introduced and some properties of them are discussed. Keywords—BCI-algebra, intuitionistic fuzzy set, intuitionistic fuzzy subalgebra with thresholds (λ, μ), intuitionistic fuzzy ideal with thresholds...

Let g be a complex semisimple Lie algebra and let r ⊂ g be any reductive Lie subalgebra such that B|r is nonsingular where B is the Killing form of g. Let Z(r) and Z(g) be, respectively, the centers of the enveloping algebras of r and g. Using a Harish-Chandra isomorphism one has a homomorphism η : Z(g) → Z(r) which, by a well-known result of H. Cartan, yields the the relative Lie algebra cohom...

We consider a class of infinite-dimensional Lie algebras which is associated to dynamical systems with invariant measures. There are two constructions of the algebras – one based on the associative cross product algebra which considered as Lie algebra and then extended with nontrivial scalar twococycle; the second description is the specification of the construction of the graded Lie algebras w...

Let U denote the quantized enveloping algebra associated to a semisimple Lie algebra. This paper studies Harish-Chandra modules for the recently constructed quantum symmetric pairs U ,B in the maximally split case. Finite-dimensional U -modules are shown to be Harish-Chandra as well as the B-unitary socle of an arbitrary module. A classification of finite-dimensional spherical modules analogous...

In her thesis [RB], Ranee Brylinski (then Gupta) studied the orbit structure of the projective variety of abelian subalgebras of a …xed dimension, k, in a simple Lie algebra, g, over C under its adjoint group, G. Fix a Borel subalgebra, b, of g and let B be the closed subgroup of G corresponding to b. Then the Borel …xed point theorem implies that the closed G-orbits are precisely the orbits of...