共查询到20条相似文献,搜索用时 15 毫秒
1.
We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi-Yetter-Drinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary Yetter-Drinfeld modules. To each braiding (a solution to the braid equation) we associate a QYD-module and the corresponding braided Heisenberg double—this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds in the braided Heisenberg double attached to the corresponding complex reflection group. 相似文献
2.
《代数通讯》2013,41(9):3029-3050
ABSTRACT Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GL n (𝕂). In particular, we define the graded quantum groups, which are Hopf π-coalgebras for π = ?[[h]] l and generalize the Drinfeld-Jimbo quantum enveloping algebras. 相似文献
3.
Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc. 相似文献
4.
IVÁN ANGIONO 《Transformation Groups》2016,21(1):1-33
We formally define and study the distinguished pre-Nichols algebra \( \tilde{B} \)(V) of a braided vector space of diagonal type V with finite-dimensional Nichols algebra B(V). The algebra \( \tilde{B} \)(V) is presented by fewer relations than B(V), so it is intermediate between the tensor algebra T(V) and B(V). Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from \( \tilde{B} \)(V) to B(V), generalizing results of De Concini and Procesi on quantum groups at roots of unity. 相似文献
5.
The structure of weak Hopf algebras corresponding to U q (sl 2) are classified by their algebra structure and coalgebra structure. The algebra structure of weak Hopf algebras corresponding to U q (sl 2) can be written as the direct sum of U q (sl 2) and an algebra of polynomials. The coalgebra structure of weak Hopf algebras corresponding to U q (sl 2) are classified by their Ext quiver. There are four types of such structures. 相似文献
6.
Finite vs affine W-algebras 总被引:1,自引:0,他引:1
In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct,
in the most general framework, the Zhu algebra ZhuΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra ZhuH V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal
enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras
(resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra ZhuH R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic
properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the
most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the
machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6
we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence
of three definitions of a finite W-algebra is established.
“I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in
his Rutgers seminar) 相似文献
7.
Jonas T. Hartwig 《Journal of Pure and Applied Algebra》2008,212(4):863-883
We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl2), Uq(sl2) and the enveloping algebra of the three-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions, and prove a Clebsch-Gordan decomposition theorem for the tensor product of two simple modules. We construct a Casimir type operator and prove that any finite-dimensional weight module is semisimple. 相似文献
8.
The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established. 相似文献
9.
We give a presentation of the Schur algebras S
Q
(2,d) by generators and relations, in fact a presentation which is compatible with Serre's presentation of the universal enveloping algebra of a simple Lie algebra. In the process we find a new basis for S
Q
(2,d), a truncated form of the usual PBW basis. We also locate the integral Schur algebra within the presented algebra as the analogue of Kostant's Z-form, and show that it has an integral basis which is a truncated version of Kostant's basis. 相似文献
10.
We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions. 相似文献
11.
该文主要考虑了拟三角Hopf代数的某种Ore -扩张问题. 对拟三角Hopf代数的Ore -扩张何时保持相同的拟三角结构给出了充分必要条件. 最后作为应用, 文章讨论了Sweedler Hopf代数和Lusztig小量子群的Ore -扩张结构. 相似文献
12.
Robert Laugwitz 《Algebras and Representation Theory》2016,19(3):547-578
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to \(\mathfrak {sl}_{2}\). 相似文献
13.
The quantized enveloping C(q)-algebra U
q
(C) associated to a Cartan matirx C is Auslander-regular and Cohen–Macaulay. This is deduced from a general theorem, which also applies to solvable polynomial algebras. The results are obtained by constructing a new filtration keeping the properties of the associated graded algebra of a given multi-filtered algebra. 相似文献
14.
Nai Hong HU Shen You WANG 《数学学报(英文版)》2014,30(10):1674-1688
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. 相似文献
15.
Zhi Xiang Wu 《数学学报(英文版)》2009,25(8):1337-1352
We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying jm = j for some integer m. We denote this algebra by wUqT(A). This algebra is a weak Hopf algebra if and only if m = 2,3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra Uq (A) of a generalized Kac-Moody algebra A. 相似文献
16.
17.
Marc Rosso 《Inventiones Mathematicae》1998,133(2):399-416
Let U
q
+ be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix. We show
that U
q
+ is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra
associated with a suitable Hopf bimodule on the group algebra of Z
n
. This method gives supersymetric as well as multiparametric versions of U
q
+ in a uniform way (for a suitable choice of the Hopf bimodule). We give a classification result about the Hopf algebras which
can be obtained in this way, under a reasonable growth condition. We also show how the general formalism allows to reconstruct
higher rank quantized enveloping algebras from U
q
sl(2) and a suitable irreducible finite dimensional representation.
Oblatum 21-III-1997 & 12-IX-1997 相似文献
18.
This note presents some results on projective modules and the Grothendieck groups K 0 and G 0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions. 相似文献
19.
The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed. 相似文献
20.
This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra. 相似文献