Schur's Double Centralizer Theorem for Triangular Hopf Algebras

2015

J. Algebra Appl.

In this paper, firstly we study the central invariant Z H (L) 0 of generalized Hom-Lie algebras L, which are monoidal Hom-Lie algebras in the category H M of H-modules for a triangular Hopf algebra (H, R). If V is an H-Hom-Lie ideal of [L, L], under certain conditions we obtain V 0 ⊆ Z H (L) 0 , where V 0 is the monoidal Hom-subalgebra of Hinvariant of V . Secondly, we describe the H-Hom-Lie ideal structure of monoidal Homalgebras in a module category.proposed the notion of Hom-associative algebras as well as were first to introduce Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras and related objects. Furthermore they obtained that a Hom-associative algebra gives rise to a Hom-Lie algebra via the commutator bracket. Wang, Kan and Chen [13] studied the structure of the generalized Lie algebras (i.e. the Lie algebras in the module category H M). Caenepeel and Goyvaerts [1] studied the Hom-Hopf algebras from a categorical view point. Motivated by this, the authors [3] extend the above work by Makhlouf and Silvestrov to the module categories. Meanwhile, we obtain same results for the generalized Hom-Lie algebras that are analogous to [13].The present paper is a continuation of [3], where we discussed generalized Hom-Lie algebras (i.e. monoidal Hom -Lie algebras in module category H M for any quasitriangular Hopf algebra H). In this paper we study the central invariant Z H (L) 0 of generalized Hom-Lie algebras L, which are monoidal Hom-Lie algebras in the category H M of H-modules for a triangular Hopf algebra (H, R). If V is an H-Hom-Lie ideal of [L, L], under certain conditions we obtain V 0 ⊆ Z H (L) 0 (Theorem 2.8), where V 0 is the monoidal Hom-subalgebra of H-invariant of V . Secondly, we describe the H-Hom-Lie ideal structures of monoidal Hom-algebras in module category H M and obtain the following results. If L is H-prime, then there exists an H-Hom-Lie ideal of [L, L] (Theorem 3.2). If L is H-simple and V is an H-Hom-Lie ideal of [L, L], then V is a solvable Hom-Lie subalgebra of L (Theorem 3.9).The paper is organized as follows. In Sec.

Section: Quasitriangular Hopf Algebramentioning

confidence: 99%

“…ad 2 x (R (2) · α(m))α 2 ([R (1) · l, y]) = 0, (2.4) for all y ∈ I, l ∈ [L, L], m ∈ L. We now consider the following two cases: …”

Section: Substituting (22) and (23) Into (21) We Obtainmentioning

confidence: 99%

Hom-ideal structure of monoidal Hom-algebras in a category

Dong

2015

J. Algebra Appl.

“…For a quasitriangular Hopf algebra H R [3], it is easy to see that the category H is a Yetter-Drinfeld category under the comodule coaction m = R 2 ⊗ R 1 · m for all m ∈ M ∈ H . In particular, if H is triangular, then the condition that R is convolution invertible means that the category H is braided symmetric.…”

Section: )mentioning

confidence: 99%

Drinfeld Double for Braided Infinitesimal Hopf Algebras

Communications in Algebra

2014

“…Furthermore, in the framework of a symmetric monoidal category, Schauenburg [14] showed that the category of Hopf bimodules is a braided monoidal category by constructing a braided monoidal category equivalence between the category of Hopf bimodules and the category of Yetter-Drinfel'd modules. On the other hand, the study of braided Lie algebras has been an interesting topic (see Bahturin et al [1], Cohen et al [6], Majid [10,11] and Wang [16,18,20]). For example, Wang [18] studied the braided Lie structures of an algebra A in the braided monoidal category of Yetter-Drinfel'd modules (see also Wang [21]).…”

Section: Introductionmentioning

confidence: 99%

On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules

Zhu

2010

Algebra Colloq.

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.