20,698 research outputs found
Harish-Chandra integrals as nilpotent integrals
Recently the correlation functions of the so-called
Itzykson-Zuber/Harish-Chandra integrals were computed (by one of the authors
and collaborators) for all classical groups using an integration formula that
relates integrals over compact groups with respect to the Haar measure and
Gaussian integrals over a maximal nilpotent Lie subalgebra of their
complexification. Since the integration formula a posteriori had the same form
for the classical series, a conjecture was formulated that such a formula
should hold for arbitrary semisimple Lie groups. We prove this conjecture using
an abstract Lie-theoretic approach.Comment: 10 page
2-matrix versus complex matrix model, integrals over the unitary group as triangular integrals
We prove that the 2-hermitean matrix model and the complex-matrix model obey
the same loop equations, and as a byproduct, we find a formula for
Itzykzon-Zuber's type integrals over the unitary group. Integrals over U(n) are
rewritten as gaussian integrals over triangular matrices and then computed
explicitly. That formula is an efficient alternative to the former
Shatashvili's formula.Comment: 29 pages, Late
Compact coalgebras, compact quantum groups and the positive antipode
In this article -that has also the intention to survey some known results in
the theory of compact quantum groups using methods different from the standard
and with a strong algebraic flavor- we consider compact o-coalgebras and Hopf
algebras. In the case of a o-Hopf algebra we present a proof of the
characterization of the compactness in terms of the existence of a positive
definite integral, and use our methods to give an elementary proof of the
uniqueness - up to conjugation by an automorphism of Hopf algebras- of the
compact involution appearing in [4]. We study the basic properties of the
positive square root of the antipode square that is a Hopf algebra automorphism
that we call the positive antipode. We use it -as well as the unitary antipode
and the Nakayama automorphism- in order to enhance our understanding of the
antipode itself
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