
SIGMA 6 (2010), 042, 18 pages arXiv:0903.5237
http://dx.doi.org/10.3842/SIGMA.2010.042
Contribution to the Special Issue “Noncommutative Spaces and Fields”
Discrete Minimal Surface Algebras
Joakim Arnlind ^{a} and Jens Hoppe ^{b}
^{a)} Institut des Hautes Études Scientifiques, Le BoisMarie, 35, Route de Chartres,
91440 BuressurYvette, France
^{b)} Eidgenössische Technische Hochschule, 8093 Zürich, Switzerland
(on leave of absence from Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden)
Received March 23, 2010, in final form May 18, 2010; Published online May 26, 2010
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 semisimple Lie algebra
providing a trivial example by itself). A special class of DMSAs
are YangMills 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.
Key words:
noncommutative surface; minimal surface; discrete Laplace operator; graph representation; matrix regularization; membrane theory; YangMills algebra.
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