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

Abstract

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...

Multiplication alteration by two-cocycles for bialgebras with weak antipode

“…Note that none of these bialgebras are Hopf algebras unless S is a group. Other more sophisticated examples are the weak quantized enveloping algebras of semisimple Lie algebras, generalized Kac-Moody algebras and superalgebras (see [1] and [6] for details).…”

Regioselective Hydroxylation of Unsymmetrical Ketones Using Cu, H2O2, and Imine Directing Groups via Formation of an Electrophilic Cupric Hydroperoxide Core

“…With the exception of the Rh-catalyzed α-C−H hydroacylation of aldimines, 8−10 the metal is proposed to coordinate to the imine N donor, which promotes the selective activation of the βand γ-C−H bonds of the most sterically hindered carbonyl substituent (i.e., functionalization occurs at substituent trans to the directing group N-substituent, see Figure 1A). 8,11,12 Inspired by the reactivity of Cu-dependent monooxygenase/ peroxygenase enzymes, 13−16 our research laboratory has published a series of papers that describe the utilization of imine directing groups, Cu, and H 2 O 2 to perform the γ hydroxylation of sp 2 and sp 3 C−H bonds and the β hydroxylation of sp 3 C−H bonds (Figure 1B; Note: following literature precedents on Cu-directed hydroxylation reactions, we named the carbonyl of the ketone C α and the adjacent position C β ). 17−20 The selectivity of these Cu-promoted oxidations, which were initially developed by Schoënecker and co-workers for the selective oxidation of steroids 21 and have been applied to the total synthesis of complex molecules, 22−24 also relies on the formation of only one of the imine isomers in unsymmetrical substrates (e.g., cyclohexyl phenyl ketone 18 ), an issue that can be avoided in the hydroxylation of symmetrical substrates (e.g., symmetrical benzophenones 18 ).…”