Related papers: Two-parameter Quantum Affine Superalgebra $U_{p,q}…
Non-Abelian fractional supersymmetry algebra in two dimensions is introduced utilizing $U_q(sl(2,\Rcc))$ at roots of unity. Its representations and the matrix elements are obtained. The dual of it is constructed and the corepresentations…
We provide a deformation, $\mathfrak{f}_{\beta}$, of Lusztig algebra $\mathbf{f}$. Various quantum algebras in literatures, including half parts of two-parameter quantum algebras, quantum superalgebras, and multi-parameter quantum…
We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups $\operatorname{GL}_n$, the two-parameter…
In this paper, we recall our renormalized quantum Q-system associated with representations of the Lie algebra $A_r$, and show that it can be viewed as a quotient of the quantum current algebra $U_q({\mathfrak n}[u,u^{-1}])\subset…
We show that some factors of the universal R-matrix generate a family of twistings for the standard Hopf structure of any quantized contragredient Lie (super)algebra of finite growth. As an application we prove that any two isomorphic…
We discuss two-parameter deformations of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). These deformations are Hopf algebras.…
We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in an earlier work for any quiver $Q$ and any one-parameter formal group $\mathbb{G}$. In this paper, we construct a comultiplication on the CoHA, making…
The two parameters quantum algebra $SU_{p,k}(2)$ can be obtained from a single parameter algebra $SU_q(2)$. This fact gives some relations between $SU_{p,k}(2)$ quantities and the corresponding ones of the $SU_q(2)$ algebra. In this paper…
Using the technique developed by Fronsdal and Galindo (Lett. Math. Phys. 27 (1993) 57) for studying the Hopf duality between the quantum algebras $Fun_{p,q}(GL(2))$ and $U_{p,q}(gl(2))$, the Hopf structure of $U_{p,q}(gl(1|1))$, dual to…
We realize the $U_q(\widehat{sl(2)})$ current algebra at arbitrary level in terms of one deformed free bosonic field and a pair of deformed parafermionic fields. It is shown that the operator product expansions of these parafermionic fields…
We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $\Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $\mathcal…
The minimal irreducible representations of $U_q[gl(m|n)]$, i.e. those irreducible representations that are also irreducible under $U_q[osp(m|n)]$ are investigated and shown to be affinizable to give irreducible representations of the…
The two parameter quantum group G_r,s is generated by five elements, four of which form a Hopf subalgebra isomorphic to GL_q(2), while the fifth generator relates G_r,s to GL_p,q(2). We construct explicitly the dual algebra of G_r,s and…
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1…
We introduce a noncommutative differential calculus on the two-parameter $h$-superplane via a contraction of the (p,q)-superplane. We manifestly show that the differential calculus is covariant under $GL_{h_1,h_2}(1| 1)$ transformations. We…
Affine transformations (dilatations and translations) are used to define a deformation of one-dimensional $N=2$ supersymmetric quantum mechanics. Resulting physical systems do not have conserved charges and degeneracies in the spectra.…
Previously we have established that the second Hamiltonian structure of the KP hierarchy is a nonlinear deformation, called $\hat{W}_{\infty}$, of the linear, centerless $W_{\infty}$ algebra. In this letter we present a free-field…
An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_3)$ is given. This result is obtained in the framework of the…
We give a simplified description of quantum affine algebras in their loop presentation. This description is related to Drinfeld's new realization via halves of vertex operators. We also define an idempotent version of the quantum affine…
We construct a Fock space representation and the action of the two-parameter quantum algebra $U_{r,s}(\frak{gl}_{\infty})$ using extended Young diagrams. In particular, we obtain an integrable representation of the two-parameter quantum…