3,213 research outputs found
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
Gauss decomposition of trigonometric R-matrices
The general formula for the universal R-matrix for quantized nontwisted
affine algebras by Khoroshkin and Tolstoy is applied for zero central charge
highest weight modules of the quantized affine algebras. It is shown how the
universal R-matrix produces the Gauss decomposition of trigonomitric R-matrix
in tensor product of these modules. Explicit calculations for the simplest case
of are presented. As a consequence new formulas for the
trigonometric R-matrix with a parameter in tensor product of -Verma
modules are obtained.Comment: 14 page
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
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