Thirty-Third Annual Victorian Algebra Conference, Sydney, Australia, 2015

Register for: VAC33

Details of talk

TitleIdentity-preserving and inverse-preserving polynomial maps from $\mathbb{C}$ to $\mathrm{SL}(n,\mathbb{C})$
PresenterAnthony Henderson (The University of Sydney)
Author(s)Dr Anthony Henderson

A map $f:\mathbb{C}\to\mathrm{SL}(n,\mathbb{C})$ is \emph{polynomial} if each
entry of $f(z)$ is a polynomial function of $z$, \emph{identity-preserving} if
$f(0)$ is the identity matrix, and \emph{inverse-preserving} if
$f(-z)=f(z)^{-1}$ for all $z$. Obviously, if $f$ is a group homomorphism then it
is both identity-preserving and inverse-preserving, but the converse is far from
true: the polynomial group homomorphisms from $\mathbb{C}$ to
$\mathrm{SL}(n,\mathbb{C})$ form a finite-dimensional algebraic variety called
the nilpotent cone, whereas the identity-preserving and inverse-preserving
polynomial maps form an infinite-dimensional variety. I will explain a natural
stratification of this infinite-dimensional variety into connected
finite-dimensional varieties, which are isomorphic to Nakajima quiver varieties
of type D.