In [1],the authors defined algebra homomorphisms from the dual RingelHall algebra of certain hereditary abelian categoryA to an appropriate q-polynomial algebra. In the case that A is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in [9]. Moreover, if the underlying graph of Q is bipartite and the matrix B associated t...

Free field constructions for the elliptic algebra A q,p (sl 2) and Baxter's eight-vertex model Abstract Three examples of free field constructions for the vertex operators of the elliptic quantum group A q,p (sl 2) are obtained. Two of these (for p 1/2 = ±q 3/2 , p 1/2 = −q 2) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 Z-al...

Free field constructions for the elliptic algebra A q,p (sl 2) and Baxter's eight-vertex model Abstract Three examples of free field constructions for the vertex operators of the elliptic quantum group A q,p (sl 2) are obtained. Two of these (for p 1/2 = ±q 3/2 , p 1/2 = −q 2) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 Z-al...

The main aim of this paper is to give classes of irreducible infinite dimensional representations and of irreducible *-representations of the q-deformed algebra U ′ q (so 2,2) which is a real form of the non-standard deformation U ′ q (so 4) of the universal enveloping algebra U (so(4, C)). These representations are described by two complex parameters and are obtained by " analytical continuati...

In this paper two types of coherent states of gl q (2)-covariant oscil-lators are investigated. 1 Quantum groups or q-deformed Lie algebra implies some specific deformations of classical Lie algebras. From a mathematical point of view, it is a non-commutative associative Hopf algebra. The structure and representation theory of quantum groups have been developed extensively by Jimbo [1] and Drin...

Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and b 6= 1, α = b − 1. The condition on α implies that Γ is formally self-dual. For b = q we use the adjacency matrix and dual adjacency matrix to obtain an action of the q-tetrahedron algebra ⊠q on the standard module of Γ. We describe four algebra homomorphisms into ⊠q from the quantum affine algebra Uq(ŝl2); using t...

The Dickson Algebra on q-variables is the algebra of invariants of the action of the mod-2 general linear group on a polynomial algebra in q-variables. We study the structure of certain ideals in this algebra as a module over the Steenrod Algebra A, and develop methods to determine which elements are hit by Steenrod operations. This allows us to display a very small set of A-generators for thes...