Weak quantum Borcherds superalgebras and their representations

Ye, Lixia; Li, Fang

doi:10.1063/1.2436732

Cited by 4 publications

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“…A natural example is the semigroup algebra of any regular monoid which is a natural generalization of group algebra. The weak quantized enveloping algebras of semi-simple Lie algebras, generalized Kac-Moody algebras and super-algebras (see [1,15,[22][23][24]) are also contained in this kind of weak Hopf algebras. The methods to construct such weak Hopf algebras in [1,15,[22][23][24] are similar, that is, replacing the group of group-like elements of the corresponding quantum enveloping algebra by some regular monoid.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It should be pointed out that in [1,15,[22][23][24], the corresponding algebras have only one center idempotent, but here are two center idempotents P and…”

Section: Introductionmentioning

confidence: 99%

“…The weak quantized enveloping algebras of semi-simple Lie algebras, generalized Kac-Moody algebras and super-algebras (see [1,15,[22][23][24]) are also contained in this kind of weak Hopf algebras. The methods to construct such weak Hopf algebras in [1,15,[22][23][24] are similar, that is, replacing the group of group-like elements of the corresponding quantum enveloping algebra by some regular monoid.Recently, some mathematical workers are interested in the generalization of U q (sl(2)) which is the simplest and most important example of quantum groups. As example, Ji and Wang [10] introduced a class of quantum algebras U q ( f (K )) and studied representations of U q ( f (K )); Wang et al [21] introduced another class of quantum algebras U q ( f (K , H )) and studied representations and the center of U q ( f (K , H )) which is a generalization of Drinfeld double of two Hopf algebras (see [21]).…”

mentioning

confidence: 99%

“…Our aim in this paper is to provide more nontrivial examples for weak Hopf algebras under the Li's meaning. So, using the similar method as in [1,15,[22][23][24], we can introduce a class of k q -algebras corresponding to U q ( f (K , H ) [1,15,[22][23][24], the corresponding algebras have only one center idempotent, but here are two center idempotents P and Q in wU From the definitions of w 1 U q and w 2 U q , we can conclude that w 2 U q is a homomorphic image of w 1 U q (see Proposition 2.9). We also study the basis of w 1 U q and w 2 U q .…”

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confidence: 99%

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Weak Hopf algebras corresponding to quantum algebras U q (f (K, H))

Hong

2012

Arab. J. Math.

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In this paper, we investigate weak Hopf algebras introduced in Li (J Algebra 208:72-100, 1998; Commun Math Phys 225:191-217, 2002) corresponding to quantum algebras U q ( f (K , H )) (see Wang et al. in Commun Algebra 30:2191-2211. A new class of algebras is defined, which is denoted by wU d q . For d = ((1, 1) | (1, 1)), denote wU d q briefly by w 1 U q ; for d = ((0, 0) | (0, 0)), denote wU d q briefly by w 2 U q . In some cases, the necessary and sufficient conditions for w 1 U q and w 2 U q to be weak Hopf algebras are given. The PBW bases of w 1 U q and w 2 U q are presented. Finally, representations and the center of w 1 U q are characterized over C with q ∈ C which is not a root of unity. Mathematics Subject Classification (2010) 16T05 · 16T20 IntroductionAlong with the introduction of quantum groups [6,11], the importance of Hopf algebras [17,20] has been widely recognized in both mathematics and physics. For example, Hopf algebra is a principal tool of studying non-commutative space [16] as quantization of commutative ones. There are some interesting generalizations of Hopf algebras. One way to generalize the concept of Hopf algebra is to introduce a kind of weak co-product such that (1) = 1 ⊗ 1 (see [4,5]). The paragroups of [18] and the face algebras of [9] are examples of this class of weak Hopf algebras. By defining a weak antipode on bialgebras, Li [13][14][15] :195-218 T is called a weak antipode of H. It should be pointed out that this concept naturally generalizes the notion of Hopf algebras, the left and right Hopf algebras [8]. There are some examples for this kind of weak Hopf algebras. A natural example is the semigroup algebra of any regular monoid which is a natural generalization of group algebra. The weak quantized enveloping algebras of semi-simple Lie algebras, generalized Kac-Moody algebras and super-algebras (see [1,15,[22][23][24]) are also contained in this kind of weak Hopf algebras. The methods to construct such weak Hopf algebras in [1,15,[22][23][24] are similar, that is, replacing the group of group-like elements of the corresponding quantum enveloping algebra by some regular monoid.Recently, some mathematical workers are interested in the generalization of U q (sl(2)) which is the simplest and most important example of quantum groups. As example, Ji and Wang [10] introduced a class of quantum algebras U q ( f (K )) and studied representations of U q ( f (K )); Wang et al. [21] introduced another class of quantum algebras U q ( f (K , H )) and studied representations and the center of U q ( f (K , H )) which is a generalization of Drinfeld double of two Hopf algebras (see [21]). Our aim in this paper is to provide more nontrivial examples for weak Hopf algebras under the Li's meaning. So, using the similar method as in [1,15,[22][23][24], we can introduce a class of k q -algebras corresponding to U q ( f (K , H )). Denote this class ofIt should be pointed out that in [1,15,[22][23][24], the corresponding algebras have only one center idempotent, but here are tw...

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It should be pointed out that in [1,15,[22][23][24], the corresponding algebras have only one center idempotent, but here are two center idempotents P and…”

Section: Introductionmentioning

confidence: 99%