Discrete Minimal Surface Algebras

TL;DR

This paper introduces discrete minimal surface algebras (DMSA) as noncommutative analogues of minimal surfaces, explores their structure, representations, and connections to membrane theory and Lie algebras.

Contribution

It defines DMSAs, proves their consistency, constructs explicit examples, and analyzes their representations using graph theory, including classification of unitary representations.

Findings

DMSAs are consistent algebraic structures related to minimal surfaces.

Explicit examples of DMSAs are constructed using semi-simple Lie algebras.

Representation graphs help classify and understand the properties of DMSA representations.

Abstract

We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d<=4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.