Host Institution:

La Trobe University

Title of Seminar:

F_q[M_n],  F_q[GL_n] and F_q[S_n] as Quantized Hyperalgebras

Speaker's Name:

Professor Zoran Rakić

Speaker's Institution:

University of Belgrade, Serbia

Time and Date:

Monday 21st November, 2011, 12:00 PM (AEDT)

Seminar Abstract:

The first part of the lecture is introductory. The notions of Hopf algebra, quantum group, quantum function algebra, quantum Frobenius morphism and some basic examples will be given. In the second part, we consider the quantum function algebra F_q[GL_n] and study the subset F_q[GL_n] of all elements of F_q[GL_n]  which are Z[q, q^-1]-valued when paired with U_q(gl_n), the unrestricted Z[q, q^-1]-integral form of U_q(gl_n), introduced by De Concini, Kac and Procesi. In particular we obtain a presentation of it by generators and relations and a PBW-like theorem. Moreover, we give a direct proof that F_q[GL_n] is a Hopf subalgebra of F_q[GL_n], and that F_q[GL_n]_|q=1= U_Z (gl_n^*). We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism from F_ε[GL_n] to F₁[GL_n] = U_Z (gl_n^*). The same analysis is done for F_q[SL_n] and (as key step) for F_q[S_n].

This talk is based on joint work with Fabio Gavarini, University Tor Vergata, Rome.

Seminar Convenor:

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