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**1 - 4**of**4**### 1 Traces in braided categories

, 2008

"... With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms RU,V are natural in the sense of [T]. We show that this con ..."

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With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms RU,V are natural in the sense of [T]. We show that this condition leads to rescaling the initial Hecke symmetry. We suggest a new way of introducing traces as properly normalized categorical morphisms End (V) → K and deduce the corresponding normalization from categorical dimensions.

### unknown title

, 2002

"... The classification of finite-dimensional triangular Hopf algebras over an algebraically ..."

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The classification of finite-dimensional triangular Hopf algebras over an algebraically

### Poisson algebras and Yang-Baxter equations

, 2009

"... Dedicated to Kazem Mahdavi, my kind mentor and friend. Abstract. We connect generalizations of Poisson algebras with the classical and associative Yang-Baxter equations. In particular, we prove that solutions of the classical Yang-Baxter equation on a vector space V are equivalent to “twisted ” Pois ..."

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Dedicated to Kazem Mahdavi, my kind mentor and friend. Abstract. We connect generalizations of Poisson algebras with the classical and associative Yang-Baxter equations. In particular, we prove that solutions of the classical Yang-Baxter equation on a vector space V are equivalent to “twisted ” Poisson algebra structures on the tensor algebra TV. Here, “twisted ” refers to working in the category of graded vector spaces equipped with Sn actions in degree n. We show that the associative Yang-Baxter equation is similarly related to the double Poisson algebras of Van den Bergh. We generalize to L∞-algebras and define “infinity ” versions of Yang-Baxter equations and double Poisson algebras. The proofs are based on the observation that Lie is essentially unique among quadratic operads having a certain distributivity property over the commutative operad; we also give an L ∞ generalization. In the appendix, we prove a generalized version of Schur-Weyl duality, which is related to the use of nonstandard Sn-module structures on V ⊗n. 1. Twisted Poisson algebras and the CYBE Throughout, we will work over a characteristic-zero field k. The tensor algebra TV = TkV satisfies the following twisted-commutativity property: each graded component V ⊗m is equipped with an Sm-module structure by permutation of components, and given homogeneous elements v, w ∈ TV of degrees |v|, |w|, we have (1.1) w ⊗ v = (21) |v|,|w | (v ⊗ w), where (21) |v|,|w | ∈ S |v|+|w | is the permutation of the two blocks {1,...,|v|}, {|v | + 1,...,|v | + |w|}. We thus say that TV is a twisted commutative algebra. 1 Similarly, we may define twisted Lie algebras. Again let A = ⊕ m≥0 Am together with an Sm action on Am for all m. A twisted Lie algebra is A together with a graded bracket { , } : A ⊗ A → A satisfying

### TRIANGULAR HOPF ALGEBRAS OVER AN ALGEBRAICALLY CLOSED FIELD OF CHARACTERISTIC 0

"... Abstract. We explain that a new theorem of Deligne on symmetric tensor categories [De2] implies, in a straightforward manner, that any finite dimensional triangular Hopf algebra over an algebraically closed field of characteristic zero has the Chevalley property, and in particular the list of finite ..."

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Abstract. We explain that a new theorem of Deligne on symmetric tensor categories [De2] implies, in a straightforward manner, that any finite dimensional triangular Hopf algebra over an algebraically closed field of characteristic zero has the Chevalley property, and in particular the list of finite dimensional triangular Hopf algebras over such a field, given in [AEG], [EG3], is complete. We also use Deligne’s theorem to settle a number of questions about triangular Hopf algebras, raised in our previous publications, and generalize Deligne’s result to nondegenerate semisimple categories in positive characteristic p, by using the lifting methods developed in [ENO].