chapter  2
54 Pages

Reconstructing Algebras

We will associate to an affine C-algebra A its affine scheme of n-dimensional representations repn A. There is a base change action by GLn on this scheme and its orbits are exactly the isomorphism classes of n-dimensional representations. We will prove the Hilbert criterium which describes the nullcone via one-parameter subgroups and apply it to prove Michael Artin’s result that the closed orbits in repn A correspond to semisimple representations. We recall the basic results on algebraic quotient varieties in geometric in-

variant theory and apply them to prove Procesi’s reconstruction result. If A ∈ [email protected], then we can recover A as

A '⇑n [trepn A] the ring of GLn-equivariant polynomial maps from the trace preserving representation scheme trepn A to Mn(C). However, the functors

[email protected]

trepn .. GL(n)-affine

do not determine an antiequivalence of categories (as they do in commutative algebraic geometry, which is the special case n = 1). We will illustrate this by calculating the rings of equivariant maps of orbit-closures of nilpotent matrices. These orbit-closures are described by the Gerstenhaber-Hesselink theorem. Later, we will be able to extend this result and study the nullcones of more general representation varieties.