Fuzzy Complex Grassmannian Spaces and their Star Products

TL;DR

This paper constructs an explicit associative star product for non-commutative complex Grassmannian spaces, generalizing previous results for projective spaces and providing a finite matrix approximation that converges to the classical algebra.

Contribution

It derives a new explicit formula for the star product on non-commutative Grassmannians, extending known results and enabling finite matrix approximations.

Findings

Explicit star product formula for complex Grassmannians.

Finite-dimensional matrix algebra approximates the commutative algebra.

Convergence to classical algebra in the infinite-dimensional limit.

Abstract

We derive an explicit expression for an associative star product on non-commutative versions of complex Grassmannian spaces, in particular for the case of complex 2-planes. Our expression is in terms of a finite sum of derivatives. This generalises previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a non-commutative algebra, represented by matrix multiplication in a finite-dimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on the complex Grassmannians.