arXiv subjects
Publications and source records attributed to Naihong Hu.
The two-parameter quantum vertex operator representation of level-one is explicitly constructed for $U_{r,s}(C^{(1)}_n)$ based on its two-parameter Drinfeld realization we give. This construction will degenerate to the one-parameter case due to Jing-Koyama-Misra (\cite{JKM2}) when $rs=1$.
In present work, we find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix algebras. Moreover, such Lie algebras generated by semi-positive definite matrices can be classified by the modified Dynkin diagrams.
In the paper, the representation rings (or the Green rings) for a family of Hopf algebras of tame type, the 2-rank Taft algebra (at $q=-1$) and its two relatives twisted by 2-cocycles are explicitly described via a representation theoretic analysis. It turns out that the Green rings can serve to detect effectively the twist-equivalent Hopf algebras here.
In the paper, we further realize the higher rank quantized universal enveloping algebra $U_q(sl_{n+1})$ as certain quantum differential operators in $\mathcal W_q(2n)$ defined over the quantum divided power algebra $\mathcal{A}_q(n)$ of rank $n$. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of $U_q(sl_{n+1})$. The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.
The purposes of this work are to construct a class of homogeneous vertex representations of $C_l^{(1)} \ (l\geq2)$, and to derive a series of product-sum identities. These identities have fine interpretation in number theory.
We put forward the exact version of two-parameter generating functions with $τ$-invariance, which allows us to give a unified and inherent definition for the Drinfeld realization of two-parameter quantum affine algebras for all the untwisted types. As verification, we first construct their level-one vertex representations of $U_{r,s}(\widehat{\mathfrak{g}})$ for simply-laced types, which in turn well-detect the effectiveness of our definitions both for $(r,s)$-generating functions and $(r,s)$-Drinfeld realization in the framework of establishing the two-parameter vertex representation theory.
The paper mainly considers the center of two-parameter quantum groups $U_{r,s}(\mathfrak{so}_{2n+1})$ via an analogue of the Harish-Chandra homomorphism. In the case when $n$ is odd, the Harish-Chandra homomorphism is not injective in general. When $n$ is even, the Harish-Chandra homomorphism is injective and the center of two-parameter quantum groups $U_{r,s}{(\mathfrak{so}_{2n+1})}$ is described, up to isomorphism.
It is proved that the entire multi-parameter (small-)quantum groups of symmetrizable Kac-Moody algebras can be realized as certain subquotients of the cotensor Hopf algebras. This is an axiomatic construction. Hopf 2-cocycle deformations variation for the construction machinery is described, moreover, the integrable irreducible modules can be constructed using this setting, even available for those in the fundamental alcove at root of unity case.
The paper explores the indecomposable submodule structures of quantum divided power algebra $\mathcal{A}_q(n)$ defined in \cite{HU} and its truncated objects $\mathcal{A}_q(n, \bold m)$. An "intertwinedly-lifting" method is established to prove the indecomposability of a module when its socle is non-simple. The Loewy filtrations are described for all homogeneous subspaces $\mathcal{A}^{(s)}_q(n)$ or $\mathcal{A}_q^{(s)}(n, \bold m)$, the Loewy layers and dimensions are determined. The rigidity of these indecomposable modules is proved. An interesting combinatorial identity is derived from our realization model for a class of indecomposable $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules. Meanwhile, the quantum Grassmann algebra $Ω_q(n)$ over $\mathcal{A}_q(n)$ is constructed, together with the quantum de Rham complex $(Ω_q(n), d^\bullet)$ via defining the appropriate $q$-differentials, and its subcomplex $(Ω_q(n,\bold m), d^\bullet)$. For the latter, the corresponding quantum de Rham cohomology modules are decomposed into the direct sum of some sign-trivial $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.
The structure of Lie algebras, Lie superalgebras and Leibniz algebras graded by finite root systems has been studied by several researchers since 1992. In this paper, we study the structure of Leibniz superalgebras graded by finite root systems, which gives an approach to study various classes of Leibniz superalgebras.
For any strong smash product algebra $A\#_{_R}B$ of two algebras $A$ and $B$ with a bijective morphism $R$ mapping from $B\ot A$ to $A\ot B$, we construct a cylindrical module $A\natural B$ whose diagonal cyclic module $Δ_{\bullet}(A\natural B)$ is graphically proven to be isomorphic to $C_{\bullet}(A\#_{_R}B)$ the cyclic module of the algebra. A spectral sequence is established to converge to the cyclic homology of $A\#_{_R}B$. Examples are provided to show how our results work. Particularly, the cyclic homology of the Pareigis' Hopf algebra is obtained in the way.
In this work, we introduce a new concept which is obtained by defining a new compatibility condition between Lie algebras and Lie coalgebras. With this terminology, we describe the interrelation between the Killing form and the adjoint representation in a new perspective.
We construct convex PBW-type Lyndon bases for two-parameter quantum groups U_{r,s}({so}_{2n+1}) with detailed commutation relations. It turns out that under a certain condition, the restricted two-parameter quantum group u_{r,s}({so}_{2n+1}) (r, s are roots of unity) is of Drinfeld double. All of Hopf isomorphisms of u_{r,s}({so}_{2n+1}), as well as u_{r,s}({sl}_n) are determined. Finally, necessary and sufficient conditions for u_{r,s}({so}_{2n+1}) to be a ribbon Hopf algebra are singled out by describing the left and right integrals.
In this article, we study the multi-parameter quantum groups defined by generators and relations associated with symmetrizable generalized Cartan matrices, together with their representations in the category $\mathcal O$. This presentation will be convenient for our later discussions. We present two explicit descriptions here: as a Hopf 2-cocycle deformation, and as the multi-parameter quantum shuffle realization of the positive part.
There are several researches on Lie algebras and Lie superalgebras graded by finite root systems. In this paper, we study Leibniz algebras graded by finite root systems and obtain some results in simply-laced cases.
We construct finite-dimensional pointed Hopf algebras \mathfrak u_{r,s}(G_2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group U_{r,s}(G_2) defined in \cite{HS}, which turn out to be of Drinfel'd doubles, where a crucial point is to give a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G_2 in 2-parameter quantum version. After furnishing possible commutation relations among quantum root vectors, we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions through determining their left and right integrals. Besides these, we determine all of the Hopf algebra isomorphisms of u_{r,s}(G_2) in terms of the description of the sets of its left (right) skew-primitive elements.
The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum $n$-space. A kind of braided category $\Cal {GB}$ of $\La$-graded $þ$-commutative associative algebras over a field $k$ is established. The quantum divided power algebra over $k$ related to the quantum $n$-space is introduced and described as a braided Hopf algebra in $\Cal {GB}$ (in terms of its 2-cocycle structure), over which the so called special $q$-derivatives are defined so that several new interesting quantum groups, especially, the quantized polynomial algebra in $n$ variables (as the quantized universal enveloping algebra of the abelian Lie algebra of dimension $n$), and the quantum group associated to the quantum $n$-space, are derived from our approach independently of using the $R$-matrix. As a verification of its validity of our discussion, the quantum divided power algebra is equipped with a structure of $U_q(\frak {sl}_n)$-module algebra via a certain $q$-differential operators realization. Particularly, one of the four kinds of roots vectors of $U_q(\frak {sl}_n)$ in the sense of Lusztig can be specified precisely under the realization.
The generalized Cartan type $\mathbf{S}$ Lie algebras in char 0 with the Lie bialgebra structures involved are quantized, where the Drinfel'd twist we used is proved to be a variation of the Jordanian twist. As the passage from char 0 to char p, their quantization integral forms are given. By the modular reduction and base changes, we obtain certain quantizations of the restricted universal enveloping algebra $\mathbf u(\mathbf{S}(n;\underline{1}))$ (for the Cartan type simple modular restricted Lie algebra $\mathbf{S}(n;\underline{1})$ of $\mathbf{S}$ type). They are new Hopf algebras of truncated $p$-polynomial noncommutative and noncocommutative deformation of dimension $p^{1+(n-1)(p^n-1)}$, which contain the well-known Radford algebra (\cite{DR}) as a Hopf subalgebra. As a by-product, we also get some Jordanian quantizations for $\mathfrak {sl}_n$, which are induced from those horizontal quantizations of $\mathbf S(n;\underline1)$.