Endomorphism algebras gure prominently in group representation theory. F or example, if G is a nite group of Lie type, the representation theory of the endomorphism algebra End G Cj G B |sometimes known as the Hecke algebra over C of G|plays a central role in unraveling the complex unipotent c haracters on G 22, 77. Another example arises in the modular representation theory of the nite general...

Given a finite quiver Q without loops, we introduce a new class of quantum algebras D(Q) which are deformations of the enveloping algebra of a Lie algebra which is a central extension of sln(Π(Q)) where Π(Q) is the preprojective algebra of Q. When Q is an affine Dynkin quiver of type A, D or E, we can relate them to Γ-deformed double current algebras. We are able to construct functors between d...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...

Four classes of three dimensional quadratic algebras of the type [Q0, Q±] = ±Q±, [Q+, Q−] = aQ0 + bQ0 + c, where (a, b, c) are constants or central elements of the algebra, are constructed using a generalization of the well known two-mode bosonic realizations of su(2) and su(1, 1). The resulting matrix representations and single variable differential operator realizations are obtained. Some rem...

The elliptic algebra Aq,p(ŝl(N)c) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the WN algebra, are constructed. The operators t(z) also close an exchange algebra when (−p1/2)NM = q−c−N for M ∈ Z. It becomes Abelian when in addition p = qNh where h is a non-zero integer. The Poisson structures...

We provide a generalization of Lieb’s triple matrix extension the Golden–Thompson inequality from algebras to setting traces on finite von Neumann algebras. More precisely, assume that ℳ is algebra equipped with tracial state τ. If 1≤p,q≤∞ 1/p+1/q=1, it shown whenever a, b, and c are self-adjoint τ-measurable operators satisfying a∈ℳ, e b ∈L p (ℳ,τ), q then following holds:

Quantum and q-deformed algebras find their application not only in mathematical physics and field theoretical context, but also in phenomenology of particle properties. We describe (i) the use of quantum algebras Uq(sun) corresponding to Lie algebras of the groups SUn, taken for flavor symmetries of hadrons, in deriving new high-accuracy hadron mass sum rules, and (ii) the use of (multimode) q-...