More Examples of Pseudosymmetric Braided Categories

Abstract

Communicated by T. H. LenaganWe study some examples of braided categories and quasitriangular Hopf algebras and decide which of them is pseudosymmetric, respectively pseudotriangular. We show also that there exists a universal pseudosymmetric braided category.Pseudosymmetric categories are a special class of braided categories and have been introduced in [10]. The motivation was the study of certain categorical structures called twines, strong twines and pure-braided structures (introduced in [1, 9, 13]). A braiding on a strict monoidal category is called pseudosymmetric if it satisfies a sort of modified braid relation; any symmetric braiding is pseudosymmetric. One of the most intriguing results obtained in [10] was that the category of Yetter-Drinfeld modules over a Hopf algebra H is pseudosymmetric if and only if H is commutative and cocommutative. We proved in [8] that pseudosymmetric categories can be used to construct representations for the group PS n = Bn [Pn,Pn] , the quotient of the braid group by the commutator subgroup of the pure braid group. There exists also a Hopf algebraic analogue of pseudosymmetric braidings: a quasitriangular structure on a Hopf algebra is called pseudotriangular if it satisfies a sort of modified quantum Yang-Baxter equation.In this paper we tie some lose ends from [8,10]. We study more examples of braided categories and quasitriangular Hopf algebras and decide when they are pseudosymmetric, respectively pseudotriangular. Namely, we prove that the canonical braiding of the category LR(H) of Yetter-Drinfeld-Long bimodules over a Hopf algebra H (introduced in [11]) is pseudosymmetric if and only if H is commutative and cocommutative. We show that any quasitriangular structure on the 4ν-dimensional Radford's Hopf algebra H ν (introduced in [12]) is pseudotriangular. We analyze the positive quasitriangular structures R(ξ, η) on a Hopf algebra with positive bases H(G; G + , G − ) (as defined in [6,7]), where ξ, η are group homomorphisms from G + to G − , and we present a list of necessary and sufficient conditions for R(ξ, η) to be pseudotriangular. If R(ξ, η) is normal (i.e. if ξ is trivial) these conditions reduce to the single relation η(uv) = η(vu) for all u, v ∈ G + .In Sec. 5 we recall the pseudosymmetric braided category PS introduced in [8] and we show that it is a universal pseudosymmetric category. More precisely, we prove that it satisfies two universality properties similar to the ones satisfied by the universal braid category B (see [5]).

Symmetries in Yetter-Drinfel’d-Long categories

“…We now give necessary and sufficient conditions for the canonical braiding of the category LR(H) to be a pseudosymmetry, we prove the necessary condition by a new method which is different from Proposition 2.5 in [10].…”

Yetter-Drinfeld-Long bimodules are modules

“…Corollary 2.4. (A, H) is an L-R-admissible pair if and only if(A, H ⊗ H * ) is an admissible pair.By the isomorphism in Theorem 2.3, we can obtain the following result in[5] directly. Proposition 2.5.…”