Yetter-drinfeld Category Double Frobenius Algebra

In this paper, biProbenius algebras in Yetter-Drinfeld categories are studied. In recent years, with the development of braided Hopf algebras, most finite dimension Hopf algebras are studied in braided categories. For biFrobenius algebras have tight relations with Hopf algebras, we discuss biFrobenius algebras in Yetter-Drinfeld categories. Especially, all the results are given in algebra system. Here, we give the basic properties of biFrobenius algebras in Yetter-Drinfeld categories, dual bases and Nakayama automorphism.In the first section, the definition of biFrobenius algebras in Yetter-Drinfeld categories is given. The dual algebras of biFrobeniua algebras in Yetter-Drinfeld categories are also biFrobeniua algebras in Yetter-Drinfeld categories. The main results are as fallowings:Proposition 1.2. Let A be a biFrobenius algebra in , for t € A, $ ?₄*, we have .Corollary 1.3. Let A be a biFrobenius algebra in, then t ?/jj, $ G /^. .Proposition 1.5. A is a biFrobenius algebra in , t ?/JJ, $ ?/^. then 1). There exists a 0 ?Alg(H, fc), and a group-like element n € //, such that V a ?A, V h e H, we have2). is the dual basis of DbF algebra A.Proposition 1.6. Let A be a biFrobeniug algebra in j^yD, for V a C A, h ?ie/J, <＃/;., thenProposition 1.7. Let a be the modular function and 5 be the modular element of A, thenProposition 1.8. If A eg yD, then ^* egTheorem 1.10. Let ,4 be a biProbeniua algebra in jj!VX>, then yl*, the dual algebra of A, is also a biFrobenius algebra in /jJyP. 1^- = e^ is the unit of A*. (.A-is the counit, the antipode is ^*- For V/, j e ^4*, Vo, 6 e ^4, the multiplication of A* is defined by m^. = (A^O'f':the comultiplkation of A* is defined by A^- (/) =In the second section, we give the definition of natural automorphism of biFrobe-nius algebra in %yD- We deduced the dual basis of biFrobenius algebras in ^ and generalized the formula of antipode in section one. The main results are:Proposition 2.2. u> : A ? u^ is the natural transformation from the functorProposition 2.3. u is the monoidal automorphism of the monoidal functorTheorem 2.4.biFrobenius algebra jTheorem 2.5. A is a biFrobenius algebra in ^yD, Va, b ?A, $ ?A*, thenIn the third section, the definition of Nakayama automorphism and CoNakayama automorphism are given. Then we introduced the properties of Nakayama automorphism and CoNakayama automorphism and the relation of Nakayama automorphism and CoNakayama automorphism with the antipode. At last, the properties of modular function and modular element are presented.Proposition 3.1. A is a biProbenius algebra in jj^Z5, V> is the antipode of A. N, CN are the Nakayama automorphism and CoNakayama automorphism respectively, for V a ?A, t 6 A, ThenTheorem 3.2. A is a biFrobenius algebra in #>T>. v-' is the antipode of A, N, CN are Nakayama automorphism and CoNakayama automorphism of v4,Va €. A, then we havecorollary 3.4. ^>, N, CN are all $yD automorphisms. Proposition 3.5. a : A 條 k(esp. g : k A) is a ^yD map.Proposition 3.6. ?- a : A A(esp, g- : A A, a go) is a % map.Theorem 3.7. N is the Nakayama automorphism of biFrobenius algebra in %yD, V a € -A, thencorollary 3.8. ?- a is the yD algebra automorphism of A and a € A* is invertable.Proposition 3.9. We have corollary 3.10. The action g- : ga is a coalgebra automorphism. g is invertable.Lemma 3.11. t are left integrals of A. Proposition 3.12. a = . And a is a jjyD algebra map.Proposition 3.13. The action *- a, *- a, a ->? a -^ are all ^^P algebra automorphisms.Proposition 3.14. 3-, ?> gag, a ?v ( ) J?a : a i ?> Q ->?a 暙- a are ^D^3 maps, and they and the antipode ^> commute with each other.