Hochschild Cohomology of Preprojective Algebras
by Pavel ETINGOF (MIT)
Abstract:
Preprojective algebras of quivers were introduced
by I. Gelfand and V. Ponomarev in 1979 as a technical tool of studying
quiver representations. More recently it became clear that these algebras
are very interesting objects by themselves. Namely, their
varieties of representations, called quiver varieties,
turned out to play a central role in geometric representation theory,
and consequently were studied in detail by many people, notably
W. Crawley-Boevey, G. Lusztig, and H. Nakajima.
I will talk about Hochschild cohomology of preprojective algebras
Π,
which was computed in a work by Crawley-Boevey, Ginzburg, and
myself (excluding the finite and affine Dynkin cases).
First of all, it turns out that the cohomology is concentrated in degrees
0,1,2, (in particular, deformations of Π are
unobstructed). Next, we show that H0(Π)= C ,
H1(Π)= C E × (L/R), and H2(Π)=L, where
$L=Π/[Π,Π]$, R is the degree zero part of Π and L
(spanned by the vertex idempotents), and E is the
derivation of L induced by the grading of Π.
The space L has a Lie algebra structure ("necklace bracket"),
which determines the bracket on H1(Π) and more generally
the structure of a Gerstenhaber algebra on the total cohomology
H*(Π). Finally, we calculate the Hilbert series of L.
The proofs are based on the theory of quiver varieties
and the theory of random unitary matrices of large size.