An Approach to Hopf Algebras via Frobenius Coordinates I

In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H \cite{Par72} and extend two recent theorems in \cite{EG}. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in \cite{FMS}. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular.Comment: 24 pages. To appear: Beitrage Alg. Geo

Algebraic Bethe Ansatz for deformed Gaudin model

The Gaudin model based on the sl_2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(\lambda) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe Ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl_2-invariant Gaudin model. As opposed to the sl_2-invariant case, the operator t(\lambda) and the Gaudin Hamiltonians are not hermitian. Finally, the inner products and norms of the Bethe states are studied.Comment: 23 pages; presentation improve

Classification of Lie bialgebras over current algebras

In the present paper we present a classification of Lie bialgebra structures on Lie algebras of type g[[u]] and g[u], where g is a simple finite dimensional Lie algebra.Comment: 26 page

A Quantum Analogue of the Bernstein Functor

We consider Knapp-Vogan Hecke algebras in the quantum group setting. This allows us to produce a quantum analogue of the Bernstein functor as a first step towards the cohomological induction for quantum groups.Comment: LaTeX2e, 16 pages; some inessential corrections have been introduce

Spherical principal series of quantum Harish-Chandra modules

The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. These modules appear in the theory of quantum bounded symmertic domains.Comment: 14 page