The various relations between q-deformed oscillators algebras and the q-deformed su(2) algebras are discussed. In particular, we exhibit the similarity of the q-deformed su(2) algebra obtained from q-oscillators via Schwinger construction and those obtained from q-Holstein-Primakoff transformation and show how the relation between su √ q (2) and Hong Yan q-oscillator can be regarded as an speci...

We shall discuss generic extension monoids associated with finite dimensional (basic) hereditary algebras of finite or cyclic type and related applications to Ringel–Hall algebras, (and hence, to quantum groups). We shall briefly review the geometric setting of quantum gln by Beilinson, Lusztig and MacPherson and its connections to Ringel– Hall algebras and q-Schur algebras. In the second part ...

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-...

In this paper, we give polyhedral realization of the crystal B(∞) of U− q (g) for the generalized Kac-Moody algebras. As applications, we give explicit descriptions of crystals for the generalized Kac-Moody algebras of rank 2, 3 and Monster Lie algebras. Introduction In his study of Conway and Norton’s Moonshine Conjecture [3] for the infinite dimensional Z-graded representation V ♮ of the Mons...

We consider algebras eiΠ (Q)ei where Π (Q) is the deformed preprojective algebra of weight λ and i is some vertex of Q in the case when Q is an extended Dynkin diagram and λ lies on the hyperplane orthogonal to the minimal positive imaginary root δ. We prove that the center of eiΠ (Q)ei is isomorphic to O (Q) which is a deformation of coordinate ring of Kleinian singularity which corresponds to...