Representation of SU(infinity) Algebra for Matrix Models

TL;DR

This paper explores how the matrix representation of SU(N) algebra converges to the Poisson algebra as N becomes large, highlighting differences between adjoint and fundamental representations.

Contribution

It clarifies the conditions under which SU(N) matrix representations approximate the SU(infinity) Poisson algebra in the large N limit.

Findings

Adjoint representation matrices approach Poisson algebra in large N limit.

Fundamental representation matrices do not approach Poisson algebra.

Provides insight into matrix model representations of infinite-dimensional algebras.

Abstract

We investigate how the matrix representation of SU(N) algebra approaches that of the Poisson algebra in the large N limit. In the adjoint representation, the (N^2-1) times (N^2-1) matrices of the SU(N) generators go to those of the Poisson algebra in the large N limit. However, it is not the case for the N times N matrices in the fundamental representation.