Skew derivations andU q (sl(2))

This note first describes the basic properties of the skew derivations on the polynomial ringk[X]. As a consequence of these properties it is proved that theq-analogue of the enveloping algebra of sl(2),U _q(sl(2)), has a unique action on C[X], where “action” means that C[X] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra ofU _q(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it. Both authors were supported by the NSF, S. Montgomery by grant DMS 87-00641, and S. P. Smith by DMS 87-02447.     

量子群U_q(sl(2 ) )的非负部分(英文)

设U≥ 0 是量子群Uq(sl(2 ) )的非负部分 .在本文中 ,我们确定了U≥ 0 的中心Z(U≥ 0 )和U≥ 0 的所有不可约表示     

Quantum anchor:U q (sl(2)) case

We introduce the tangent spaceT(H _q ) on the quantum hyperboloid (A _0,q ^c ) and equip it with an action onA _0,q ^c being a deformation of the action of vectors fields on functions. An embeddingsl(2) _q →T(H _q ) ofq-deformed Lie algebrasl(2) being an analogue of the anchorsl(2)→Vect (H) is called “quantum anchor”.     

WEIGHT PROPERTY FOR IDEALS OF U q (sl(2))

Let k be an arbitrary field of characteristic zero, and U be the quantized enveloping algebra U _q (sl(2)) over k. The aim of this present paper is to study the ideals of U at q not a root of unity. It turns out that every non-zero ideal of U can be generated by at most two highest weight vectors under the adjoint action, and by a sum of two highest weight vectors. This weight property make it possible to give a complete list of all prime (primitive, maximal) ideals of U according to their generators.     

量子群Uq（sl（2））的非负部分

设U≥0是量子群Uq(sl(2))的非负部分,在本中,我们确定了U≥0的中心Z（U≥0）和U≥0有不可约表示。     

The embedding ofU q(sl (2)) and sine algebras in generalized Clifford algebras

In this note, we establish the connection between certain quantum algebras and generalized Clifford algebras (GCA). Precisely, we embed the quantum tori Lie algebra andU _q(sl (2)) in GCA.     

On the bosonization ofL-operators for the quantum affine algebraU q (sl 2)

Some relations between different objects associated with quantum affine algebras are reviewed. It is shown that the Frenkel-Jing bosonization of a new realization of the quantum affine algebra as well as bosonization of L-operators for this algebra can be obtained from Zamolodchikov-Faddeev algebras defined by the quantum R-matrix satisfying unitarity and crossing-symmetry conditions.On leave of absence from the ITP, Kiev 252143, Ukraine. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 64–77, July, 1995.     

On the Indecomposable Modules of Uq（sl（2）） Constructed via Ore-extension

Let A be a subalgebra of Uq (sl(2)) generated by K, K-1 and F and Aδ be a subalgebra of Uq (sl(2)) generated by K, K-1 (and also Fd if q is a primitive d-th root of unity with d an odd number). Given an Aδ -module M, a Uq (sl(2))-module AAδ M is constructed via the iterated Ore extension of Uq (sl(2)) in a unified framework for any q. Then all the submodules of AAδ M are determined for a fixed finite-dimensional indecomposable Aδ -module M . It turns out that for some indecomposable Aδ -module M , the Uq (sl(2))-module AAδ M is indecomposable, which is not in the BGG-categories Oq associated with quantum groups in general.     

Quantum superalgebras u q (sl(m|n)) at roots of unity

Finite dimensional Hopf superalgebras u _q (sl(m|n)) corresponding to the Lie superalgebras sl(m|n) are constructed. The PBW type basis and the left and right integrals of u _q (sl(m|n)) are obtained. Furthermore, the group of Hopf superalgebra automorphisms is described.     

Generic modules over the quantum groupU t (sl(2)) att a root of unit

LetA be a finite dimensionalk-algebra, an indecomposable (left)A-moduleM is called generic providedM is infinitek-dimensional but finite length as (right) End _A M-module. In this paper we given the explicit structure of generic modules and their endomorphism rings over the finite dimensional quantum groupU _t (sl(2)) att being a root of unit.     

量子群V_q(sl(2))的商代数

本文研究当q是单位根时,V_q(sl(2))在关系H~r=K~r=1,E~(mr)=F~(nr)=0下的商代数V_q(m,n)的构造与分解,以及它的区块结构.为此,首先将U_q(sl(2))的基本性质和重要结论推广到V_q(sl(2)),并研究V_q(sl(2))的模的基本性质.利用这些结论,我们逐步构造出V_q(m,n)的左理想,并将V_q(m,n)分解成不可分解的左理想的直和.然后,把V_q(m,n)的不可分解的左理想合并成区块,并研究区块结构,从而把V_q(m,n)的表示问题归结成一个代数表示论的问题.     

Constructing indecomposable representations over quantum group ${\cal U}_q(sl(2))$

In this article, we discuss the infinite dimensional indecomposable Harish-Chandra representations over ${\cal U}_q(sl(2))$. As an application, we answer a question of Ringel on the annihilator ideals of simple modules.     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

The cartan-weyl basis for the yangian doubleDY (sl 3)

Using the analogues of Cartan-Weyl elements, we construct a realization of the Yangian Y (sl₃). This enables us to derive the explicit comultiplication formulas in Y (sl₃), the quantum double DY (sl₃) of the Yangian, and the universal R-matrix. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 423–438, June, 1997.     

Matching Realization of Uq（sln＋1） of Higher Rank in the Quantum Weyl Algebra Wq（2n）

In the paper, we further realize the higher rank quantized universal enveloping algebra Uq（sln＋1） as certain quantum differential operators in the quantum Weyl algebra Wq （2n） defined over the quantum divided power algebra Sq（n） of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of Uq（sln＋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.     

Equation f(p(x)) = q(f(x)) for given real functions p, q

We investigate functional equations f(p(x)) = q(f(x)) where p and q are given real functions defined on the set ? of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions p, q which are strictly increasing and continuous on ?. In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction of any solution of this equation if some exists. This construction is demonstrated in detail and discussed by means of an example.     

Cocycles on a finite dimensional quotient of uq (sl(2))

We prove that certain algebra quotients of Hopf algebras are twisted Hopf algebras. On the other handu_q (sl(2)) is a crossed product of a central subalgebra with a quotient [Ubar], when q is a root of 1. Using the cocycle involved in this crossed product we construct non-trivial complex cocycles τ and we find the isomorphism classes of the corresponding twisted Hopf algebras _τ [Ubar]. These provide complex projective representations of [Ubar] which are not ordinary representations.     

量子群u_q(sl(2))中理想的唯一分解

本文利用量子群uq(sl(2))的表示及其理想的生成子,证明了uq(sl(2))的任一非零理想在某种意义下可唯一分解为若干个素理想的乘积.由此得到理想的求根公式.