A deformation of the harmonic oscillator algebra associated with the Morse potential and the SU (2) algebra is derived using the quantum analogue of the anharmonic oscillator. We use the quantum os-cillator algebra or q-boson algebra which is a generalisation of the Heisenberg-Weyl algebra obtained by introducing a deformation parameter q. Further, we present a new algebraic realization of the ...

The origin of this paper is in a question that was asked of the author by Michael Wellman, a computer scientist who works in artificial intelligence at Wright Patterson Air Force Base in Dayton, Ohio. He wanted to know if, starting with R and its usual topology and product partial order, he could linearly reorder every finite subset and still obtain a continuous function from R into R that was ...

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...

Frenkel-Reshetikhin introduced q-characters of finite dimensional representations of quantum affine algebras [6]. We give a combinatorial algorithm to compute them for all simple modules. Our tool is t-analogue of the q-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition. We need the theory of quiver varieties for the definition...

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 propose a way to separate variables in rational integrable $\mathfrak{gl}(n)$ spin chain with an arbitrary finite-dimensional irreducible representation at each site and generic twisted periodic boundary conditions. Firstly, we construct basis that diagonalises higher-rank version of the Sklyanin B-operator; construction is based on recursive usage embedding $\mathfrak{gl}(k)$ into $\mathfra...

Recently B. Hartwig and the second author found a presentation for the three-point sl2 loop algebra via generators and relations. To obtain this presentation they defined an algebra ⊠ by generators and relations, and displayed an isomorphism from ⊠ to the three-point sl2 loop algebra. We introduce a quantum analog of ⊠ which we call ⊠q. We define ⊠q via generators and relations. We show how ⊠q ...

We show that every Lie algebra or superLie algebra has a canonical braiding on it, and that in terms of this its enveloping algebra appears as a flat space with braided-commuting coordinate functions. This also gives a new point of view about q-Minkowski space which arises in a similar way as the enveloping algebra of the braided Lie algebra gl2,q. Our point of view fixes the signature of the m...