1. Lecture 1: Affine Lie algebras and the Fock representation of ĝln. 1.1. The loop algebra construction. Let g be a complex reductive Lie algebra and let L denote the algebra of Laurent polynomials in one variable L = C[t, t−1]. The loop algebra over g is L(g) = L ⊗ g, which is a Lie algebra with the bracket [t ⊗ x, t ⊗ y]0 = t[x, y]. (1.1) The elements of the loop algebra may be regarded as r...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If A is a reflexive, amenable Banach algebra such that for each maximal left ideal L of A (i) the quotient A/L has t...

A bstract Recently, Gaiotto and Rapcak proposed a generalization of W N algebra by considering the symmetry at corner brane intersection (corner vertex operator algebra). The algebra, denoted as Y L,M,N , is characterized three non-negative integers L, M, . It has manifest triality automorphism which interchanges can be obtained reduction 1+∞ with “pit” in plane partition representation. Later,...

The Birman–Murakami–Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free over Z[δ±1, l±1]/(m(1− δ)− (l− l−1)) of rank (2n + 1)n!! − (2n−1 + 1)n!, where n!! = 1 · 3 · · · (2n − 1). We also show it is a cellular algebra over suitable rings. The Brauer algebra of type Dn is a homomorphic ring image and is also semisimple and free of the same rank, but over the ring Z[δ±1]. A r...

A commutative power-associative algebra A of characteristic >5 with an idempotent u may be written1 as the supplementary sum ^=^4„(l)+4u(l/2)+^4u(0) where 4U(X) is the set of all xx in A with the property xx« =Xxx. The subspaces Au(l) and .4K(0) are orthogonal subalgebras, [AU(1/2)]2QAU(1)+AU(0) andAu(K)Au(l/2) C4„(l/2)+^4u(l—X) forX=0, 1. The algebra A is called w-stable if 4u(X)-4„(l/2)C.4u(l...

We define the Hopf algebra structure on the Grothendieck group of finitedimensional polynomial representations of Uq ĝlN in the limit N → ∞. The resulting Hopf algebra Rep Uq ĝl∞ is a tensor product of its Hopf subalgebras Repa Uqĝl∞, a ∈ C/q. When q is generic (resp., q is a primitive root of unity of order l), we construct an isomorphism between the Hopf algebra Repa Uq ĝl∞ and the algebra of...

The purpose of this article is to develop the notions of amenabilityfor vector valued group algebras. We prove that L1(G, A) is approximatelyweakly amenable where A is a unital separable Banach algebra. We givenecessary and sufficient conditions for the existence of a left invariant meanon L∞(G, A∗), LUC(G, A∗), WAP(G, A∗) and C0(G, A∗).

Effect algebras are the main object of study in quantum mechanics. Module measures those defined on an effect algebra with values a topological module. Let R be ring and M R-module. L algebra. The range module measure ?:L?M is studied. Among other results, we prove that if sRDP ?-effect natural basis ?:L?R countably additive measure, then ? has bounded variation.