128 research outputs found
Curvature and geometric modules of noncommutative spheres and tori
When considered as submanifolds of Euclidean space, the Riemannian geometry
of the round sphere and the Clifford torus may be formulated in terms of
Poisson algebraic expressions involving the embedding coordinates, and a
central object is the projection operator, projecting tangent vectors in the
ambient space onto the tangent space of the submanifold. In this note, we point
out that there exist noncommutative analogues of these projection operators,
which implies a very natural definition of noncommutative tangent spaces as
particular projective modules. These modules carry an induced connection from
Euclidean space, and we compute its scalar curvature
Matrix regularization of embedded 4-manifolds
We consider products of two 2-manifolds such as S^2 x S^2, embedded in
Euclidean space and show that the corresponding 4-volume preserving
diffeomorphism algebra can be approximated by a tensor product SU(N)xSU(N) i.e.
functions on a manifold are approximated by the Kronecker product of two SU(N)
matrices. A regularization of the 4-sphere is also performed by constructing
N^2 x N^2 matrix representations of the 4-algebra (and as a byproduct of the
3-algebra which makes the regularization of S^3 also possible).Comment: 22 pages, v2: published version, minor correction
Discrete Minimal Surface Algebras
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
A noncommutative catenoid
A noncommutative algebra corresponding to the classical catenoid is
introduced together with a differential calculus of derivations. We prove that
there exists a unique metric and torsion-free connection that is compatible
with the complex structure, and the curvature is explicitly calculated. A
noncommutative analogue of the fact that the catenoid is a minimal surface is
studied by constructing a Laplace operator from the connection and showing that
the embedding coordinates are harmonic. Furthermore, an integral is defined and
the total curvature is computed. Finally, classes of left and right modules are
introduced together with constant curvature connections, and bimodule
compatibility conditions are discussed in detail
Representation theory of C-algebras for a higher order class of spheres and tori
We construct C-algebras for a class of surfaces that are inverse images of
certain polynomials of arbitrary degree. By using the directed graph associated
to a matrix, the representation theory can be understood in terms of ``loop''
and ``string'' representations, which are closely related to the dynamics of an
iterated map in the plane. As a particular class of algebras we introduce the
``Henon algebras'', for which the dynamical map is a generalized Henon map, and
give an example where irreducible representations of all dimensions exist.Comment: 14 page
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