Hall 123

Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

Ajay Ramadoss

Indiana University, Bloomington

We identify the representation homologies of an augmented algebra
$A$ over a field $k$ (of characteristic $0$) with the Chevalley-Eilenberg
homologies of Lie coalgebras that are naturally related to differential
graded coalgebras Koszul dual to $A$. As a consequence of this
identification, one obtains a derived Harish-Chandra homorphism relating
the $n$-th representation homology of $A$ to the $n$-th symmetric power of its
first representation homology. When $A=k[x,y]$, we conjecture that this
derived Harish-Chandra homomorphism is an isomorphism. As evidence for our
conjecture, we have been able to show that our conjecture is true after
passage to an appropriate limit. We have also proven (through a different
approach) a Macdonald type combinatorial identity that follows directly
from our conjecture. This is a joint work in progress with Yuri Berest, Giovanni Felder, Aliaksandr
Patotski and Thomas Willwacher.