Noncommutative complete intersections and matrix integrals

TL;DR

This paper introduces the class of representation complete intersections (RCI) in noncommutative algebras, proves their relation to noncommutative complete intersections (NCCI), and derives explicit formulas for their homology groups using matrix integrals.

Contribution

It defines RCI algebras, establishes their equivalence with NCCI algebras, and provides explicit homology formulas via noncommutative cyclic Koszul complexes and matrix integrals.

Findings

RCI algebras include those from quivers.

Any graded RCI algebra is NCCI.

Explicit Hilbert series formulas for cyclic and Hochschild homology.

Abstract

We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Rep_n(A), of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is another interesting class of associative algebras called noncommutative complete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also obtain explicit formulas for the Hilbert series of each nonvanishing cyclic and Hochschild homology group of an RCI algebra. The proof involves a noncommutative cyclic Koszul complex, K(A), and a matrix integral similar to the one arising in quiver gauge theory.