In this paper we discuss a relationship between the following two algebras: (i) the subconstituent algebra T of a distance-regular graph that has q-Racah type; (ii) the q-tetrahedron algebra ⊠q which is a q-deformation of the three-point sl2 loop algebra. Assuming that every irreducible T -module is thin, we display an algebra homomorphism from ⊠q into T and show that T is generated by the imag...

We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

In this thesis, we investigate two ways of generalising the preprojective algebra. First, we introduce the multiplicative preprojective algebra, Λ(Q), which is a multiplicative analogue of the deformed preprojective algebra, introduced by Crawley-Boevey and Holland. The special case q = 1 is the undeformed multiplicative preprojective algebra, which is an analogue of the ordinary (undeformed) p...

We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

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 ...

We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra Bn(−q , q) and the quantum algebra associated to the symplectic Lie algebra sp2m. In particular, we deduce that this Schur–Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang ([37]) in the symplectic case. As a byproduct, we show tha...

We introduce a new algebra Bl(z, q) depending on two nonzero complex parameters such that Bl(q n , q) at q = 1 coincides with the Brauer algebra Bl(n). We establish an analog of the Brauer–Schur–Weyl duality where the action of the new algebra commutes with the representation of the twisted deformation Uq(on) of the enveloping algebra U(on) in the tensor power of the vector representation.

The quantum algebra su q (2) is introduced as a deformation of the ordinary Lie algebra su(2). This is achieved in a simple way by making use of q-bosons. In connection with the quantum algebra su q (2), we discuss the q-analogues of the harmonic oscillator and the angular momentum. We also introduce q-analogues of the hydrogen atom by means of a q-deformation of the Pauli equations and of the ...

Let V m,n p,q be the irreducible representation of the Virasoro algebra L with central charge cp,q = 1− 6(p − q) /pq and highest weight hm,n = [(np−mq)−(p−q)]/4pq, where p, q > 1 are relatively prime integers, and m, n are integers, such that 0 < m < p, 0 < n < q. For fixed p and q the representations V m,n p,q form the (p, q) minimal model of the Virasoro algebra [1]. For N > 0 let LN be the L...

It is well-known that the Macfarlane-Biedenharn q-oscillator and its generalization has no Hopf structure, whereas the Hong Yan q-oscillator can be endowed with a Hopf structure. In this letter, we demonstrate that it is possible to construct a general q-oscillator algebra which includes the Macfarlane-Biedenharn oscillator algebra and the Hong Yan oscillator algebra as special cases. E-mail ad...