Let V be an affine algebraic variety over a commutative field K and let A be the K-algebra of regular (polynomial) functions on V.
The group of automorphisms of V, namely Aut_K (A), is an elusive object in Mathematics. For a curve V, it has been shown by Hurwitz (1893) that Aut_K(A) is a finite group, except for some specific genus zero curves. On the opposite, for
the affine plane V=K^2, the group Aut_K(A) is an "infinite dimensional algebraic" group. In general, the group Aut_K(V) is not linear, i.e. it cannot be embedded in some linear group GL_n(L), even if the integer n and the field L are very big.
After a general introduction, we will discuss the following two questions:
which properties of linear groups extend to Aut_K(A), and
which properties of finite dimensional Lie algebras extend to the Lie algebra Der_K(A) of vector fields on V?
In an ongoing collaboration with O. Bezushchak and A. Petravchuk from Kiev University, we investigate these questions in the following more general setting.
First it can be assumed that K is a commutative algebra instead of a field, or, equivalently that V is a family of affine algebraic varieties. Second, we can assume that A is PI algebra. It means that A is an associative algebra satisfying some polynomial identity, usually different from
the identity XY=YX defining the commutative algebras. Roughly speaking the PI algebras occur when the algebra of functions on V is replaced with the algebra of the endomorphisms of some bundle on V. This is one of the standard approaches to non-commutative geometry.
In this collaboration, we had established analogs of classical theorems of Selberg, Burnside,
and Schur for Aut_K(A). Also we have proved an analog of the Engel theorem for Der_K(A).
Christophe Garban
