arXiv subjects
Publications and source records attributed to Phung Ho Hai.
In this work we study some properties of comldules over (non-cosemisimple) Hopf algebras possessing integrals, which are also called co-Frobenius Hopf algebras. We apply the result obtained to the classification of representations of quantum groups of type $A_{0|0}$.
In this paper we construct and study the representation theory of a Hopf C^*-algebra with approximate unit, which constitutes quantum analogue of a compact group C^*-algebra. The construction is done by first introducing a convolution-product on an arbitrary Hopf algebra H with integral, and then constructing the L_2 and C^*-envelopes of H (with the new convolution-product) when H is a compact Hopf *-algebra.
We show that a quantum super matrix in standard format is invertible if and only if its block matrices of even entries are invertible. We prove the q-analog of the well-known formula for the Berezinian.
Given a Hecke symmetry $R$, one can define a matrix bialgebra $E_R$ and a matrix Hopf algebra $H_R$, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to $R$. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and that the fusion coefficients of simple representations depend only on the rank of the Hecke symmetry. Further we compute the quantum rank of simple representations. We also show that the quantum semi-group is ``Zariski'' dense in the quantum group. Finally we give a formula for the integral.
We introduce the notion of characters of comodules over coribbon Hopf algebras. The case of quantum groups of type $A_n$ is studied. We establish a characteristic equation for the quantum matrix and a q-analogue of Harish-Chandra- Itzykson-Zuber integral
Bialgebras associated to Yang-Baxter operators satisfying the Hecke equation, are considered. It is shown that they are Koszul algebras. Their Poincare' series are calculated via the Poincare' series of the corresponding quantum spaces.
We study the Poincaré series of the quantum spaces associated to a Hecke operator, i.e., a Yang-Baxter operator satisfying the equation $(x+1)(x-q)=0$. The Poincaré series of the corresponding matrix bialgebra is also considered. Using an old result on Polyá frequency sequence, we show that the Poincaré series of quantum spaces are always rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be rational functions having negative roots and positive poles. In particular, we show that the rank of an even Hecke operator should be greater than the dimension of the vector space it is acting on.
The dimension of the third homogeneous component of a matrix quantum bialgebra, determined by pair of quantum spaces, is calculated. The Poincaré series of some deformations of $GL(n)$ is calculated. A new deformation of $GL(3)$ with the correct dimension is given.