相关论文: The quantum determinant of the elliptic algebra $\…
In the Hopf algebra GL_{p,q}(2) the determinant is central iff p=q. In this case we put determinant to be equal to 1 to get SL_q(2). In this paper I consider the case when p/q is a root of unity; and, consequently, a power of the…
We reformurate a central extension of Felder's elliptic quantum group in the FRST formulation as a topological algebra E_{q,p}(gl_N) over the ring of formal power series in p. We then discuss the isomorphism between E_{q,p}(gl_N) and the…
We show how to construct central and grouplike quantum determinants for FRT algebras A(R). As an application of the general construction we give a quantum determinant for the q-Lorentz group.
We give the center of the elliptic quantum group in general case. Based on the Dynamic Yang-Baxter Relation and the fusion method, we prove that the center commute with all generators of the elliptic quantum group. Then for a kind of…
We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). The…
In this paper we generalize certain results concerning quantum affine algebra $U_{q}(\hat{sl_{2}})$ at the critical level to the corresponding elliptic case $E_{q,p}(\hat{sl_2})$. Using the Wakimoto realization of the algebra…
As a particular one parameter deformation of the quantum determinant, we introduce a quantum $\alpha$-determinant and study the $\mathcal{U}_q(\mathfrak{gl}_n)$-cyclic module generated by it: We show that the multiplicity of each…
We present a direct construction of abstract generators for q-deformed W_N algebras. This procedure hinges upon a twisted trace formula for the elliptic algebra A_{q,p}(sl(N)_c) generalizing the previously known formulae for quantum groups.
Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, F_ell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter…
We determine the center of a localization of ${\mathcal U}_q({\mathfrak n}_\omega)\subseteq {\mathcal U}^+_q({\mathfrak g})$ by the covariant elements (non-mutable elements) by means of constructions and results from quantum cluster…
We discuss some aspects of the deformed W-algebras W_{q,t}[g]. In particular, we derive an explicit formula for the Kac determinant, and discuss the center when t^2 is a primitive k-th root of unity. The relation of the structure of…
The quantum $\alpha$-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic $\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra $\mathcal{A}_q(\mathrm{Mat}_2)$…
After reviewing the recent results on the Drinfeld realization of the face type elliptic quantum group B_{q,lambda}(sl_N^) by the elliptic algebra U_{q,p}(sl_N^), we investigate a fusion of the vertex operators of U_{q,p}(sl_N^). The basic…
The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a…
For a slightly generalised version of the Jimbo quantum group associated with any finite dimensional simple Lie algebra $\mathfrak{g}$, we show that its centre is a polynomial algebra. We construct a set of algebraically independent central…
We consider the affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in the Lie algebra $\mathfrak{gl}_N$. Due to a recent result of Arakawa and Premet, the center of this vertex algebra is an…
It is shown that the elliptic algebra ${\cal A}_{q,p}(\hat{sl}(2)_c)$ at the critical level c=-2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and…
A complex elliptic curve $E$ can be defined as the quotient of the analytic space $\mathbb{C}^*$ by a discrete action of the cyclic group $q^{\mathbb{Z}}$ for $\vert q\vert \neq 1$. We study the boundary case when $\vert q\vert =1$, which…
The scaling limit $A_{\hbar,\eta}(\hat{sl_2})$ of the elliptic algebra $A_{q,p}(\hat{sl_2})$ is investigated. The limiting algebra is defined in terms of a continuous family of generators being Fourier harmonics of Gauss coordinates of the…
We present a systematic derivation of the abelianity conditions for the $q$-deformed $W$-algebras constructed from the elliptic quantum algebra $\mathcal{A}_{q,p}\big(\widehat{\mathfrak{gl}}(N)_{c}\big)$. We identify two sets of conditions…