Related papers: Explicit Central Elements of $U_q(\mathfrak{gl}(N+…

When the parameter of deformation q is a m-th root of unity, the centre of U_q(sl(N))$ contains, besides the usual q-deformed Casimirs, a set of new generators, which are basically the m-th powers of all the Cartan generators of U_q(sl(N)).…

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view…

We express the comultiplication of the generators in Drinfelds second realization of the quantum affine algebra U_q(sl_2^), induced by the comultiplication of the generators in the Drinfeld-Jimbo realization of U_q(sl_2^) in terms of…

The connection between the R-matrix realization and Drinfeld's realization of the quantum loop algebra $U_q(D^{(2)}_n)$ is considered using the Gaussian decomposition approach proposed by J. Ding and I. B. Frenkel. Our main result is a…

Following the approach of Ding and Frenkel [Comm. Math. Phys. 156 (1993), 277-300] for type $A$, we showed in our previous work [J. Math. Phys. 61 (2020), 031701, 41 pages] that the Gauss decomposition of the generator matrix in the…

A level-one representation of the quantum affine superalgebra $\U_q(\hat{\frak{sl}}(M+1|N+1))$ and vertex operators associated with the fundamental representations are constructed in terms of free bosonic fields. Character formulas of…

The construction of a generic representation of $g\ell(n+1)$ or of the trigonomentric deformation of its enveloping algebra known as algebraic induction is conveniently formulated in term of Lax matrices. The Lax matrix of the constructed…

In this article we construct a large family of $R$-matrices for various extensions of small quantum groups by grouplike elements. The extensions are in correspondence to lattices between root and weight lattice and admit $R$-matrices in…

This paper reveals some new structural property for the $i$-quantum group U^i(n) and constructs a certain hyperalgebra from the new structure which has connections to finite symplectic groups at the modular representation level.

Quantum superalgebras $su_{q}(m\mid n)$ are studied in the framework of $R$-matrix formalism. Explicit parametrization of $L^{(+)}$ and $L^{(-)}$ matrices in terms of $su_{q}(m\mid n)$ generators are presented. We also show that quantum…

The Universal T-matrix is the capstone of the structure that consists of a quantum group and its dual, and the central object from which spring the T-matrices (monodromies) of all the associated integrable models. A closed expression is…

The quantum superalgebra $U_q[gl(2/1)]$ is given as both a Drinfel'd--Jimbo deformation of $U[gl(2/1)]$ and a Hopf superalgebra. Finite--dimensional representations of this quantum superalgebra are constructed and investigated in a basis of…

Let $\mathfrak{g}'\subset \mathfrak{g}$ be the pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces $\mathbb{C}^{N-2}\subset \mathbb{C}^N$ and $U_q(\mathfrak{g}')\subset…

This paper concerns the positive part $U^+_q$ of the quantum group $U_q({\widehat{\mathfrak{sl}}}_2)$. The algebra $U^+_q$ has a presentation involving two generators that satisfy the cubic $q$-Serre relations. We recently introduced an…

Let $\g$ be a complex orthogonal or symplectic Lie algebra and $\g'\subset \g$ the Lie subalgebra of rank $\rk \g'=\rk \g-1$ of the same type. We give an explicit construction of generators of the Mickelsson algebra $Z_q(\g,\g')$ in terms…

The Holstein-Primakoff and the Dyson realizations of the Lie superalgebra $gl(n/m)$ are generalized to the class of the quantum superalgebras $U_q[gl(n/m)]$ for any $n$ and $m$. It is shown how the elements of $U_q[gl(n/m)]$ can be…

Let $Y_{M|N}(\mathfrak{s})$ be the super Yangian associated with an arbitrary fixed $0^M1^N$-sequence $\mathfrak{s}$. In the present paper, we give a new formula for the quantum Berezinian by using the parabolic generators, which…

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,q)), q prime-power, by applying a Verlinde-like formula on the…

This thesis provides a partial answer to a question posed by Greg Kuperberg in q-alg/9712003 and again by Justin Roberts as problem 12.18 in "Problems on invariants of knots and 3-manifolds", math.GT/0406190, essentially: "Can one describe…

We provide a Faddeev-Reshetikhin-Takhtajan's RTT approach to the quantum group Fun(GL_{r,s}(n)) and the quantum enveloping algebra U_{r,s}(gl_n) corresponding to the two-parameter R-matrix. We prove that the quantum determinant det_{r,s}T…