Description
The ring of invariant polynomial functions on the variety
of N x N matrices is generated by the elementary symmetric functions
on the eigenvalues of the matrix. These generators feature as the
coefficients in the Cayley-Hamilton identity, a canonical linear
relation amongst the first N powers of the matrix. When we move from
the classical group to its quantization, a Cayley-Hamilton identity
persists, and once again the coefficients give canonical generators
for the ring of invariants in the quantum coordinate ring $O_q(GL_N)$,
and these in turn give rise to generators for the center of $U_q(gl_N)$.
In this talk, I will recall the two well-known, distinct,
quantizations -- FRT algebras and reflection equation algebras --
which deserve the name $O_q(GL_N)$: they quantize two different, equally
canonical Poisson brackets on G. Then I will explain remarkably simple
combinatorial formulas for the Cayley-Hamilton generators in each
case, and how the two cases relate to one another. For the FRT
algebras, this is work of Domokos-Lenagan, and for reflection equation
algebras, this is my joint work with Noah White, which builds from
Domokos-Lenagan.