Shouchuan Zhang

DOUBLE BICROSSPRODUCTS IN BRAIDED TENSOR CATEGORIES

The double bicrossproduct D = A φ α ⋈ψ β H of two bialgebras A and H is constructed in a braided tensor category and the necessary and sufficient conditions for D to be a bialgebra are given. The universal property of double bicrossproduct is obtained.

Braided m-Lie Algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of EndFM, where M is a Yetter–Drinfeld module over B with dimB < ∞. In particular, generalized classical braided m-Lie algebras slq, f(GMG(A), F) and ospq, t (GMG(A), M, F) of generalized matrix algebra GMG(A) are constructed and their connection with special generalized matrix Lie superalgebra sls, f(GMZ_2(As), F) and orthosymplectic generalized matrix Lie super algebra osps, t (GMZ_2(As), Ms, F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.

Classification of PM Quiver Hopf Algebras

We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter–Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field k is the complex field and G is a finite abelian group, we classify quiver Hopf algebras over G, multiple Taft algebras over G and Nichols algebras in . We show that the quantum enveloping algebra of a complex semisimple Lie algebra is a quotient of a semi-path Hopf algebra.

CLASSIFICATION OF QUIVER HOPF ALGEBRAS AND POINTED HOPF ALGEBRAS OF TYPE ONE

Quiver Hopf algebras are classified by means of ramification system with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one. 10.1017/S0004972712000494

Pointed Hopf algebras with classical Weyl groups

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊n supported by 𝕊n are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊n supported by A to be finite dimensional.

Local Quasitriangular Hopf Algebras

We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. The category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.