48,769 research outputs found
The spectra of the spherical and euclidean triangle groups
We derive the spectrum of the Laplace-Beltrami operator on the quotient
orbifold of the non hyperbolic triangle groups.Comment: accepted in the Journal of the Australian Mathematical Societ
Framization of the Temperley-Lieb Algebra
We propose a framization of the Temperley-Lieb algebra. The framization is a
procedure that can briefly be described as the adding of framing to a known
knot algebra in a way that is both algebraically consistent and topologically
meaningful. Our framization of the Temperley-Lieb algebra is defined as a
quotient of the Yokonuma-Hecke algebra. The main theorem provides necessary and
sufficient conditions for the Markov trace defined on the Yokonuma-Hecke
algebra to pass through to the quotient algebra. Using this we construct
1-variable invariants for classical knots and links, which, as we show, are not
topologically equivalent to the Jones polynomial.Comment: 30 page
Homogeneous manifolds from noncommutative measure spaces
Let M be a finite von Neumann algebra with a faithful trace . In this
paper we study metric geometry of homogeneous spaces O of the unitary group U
of M, endowed with a Finsler quotient metric induced by the p-norms of ,
, . The main results include the following.
The unitary group carries on a rectifiable distance d_p induced by measuring
the length of curves with the p-norm. If we identify O as a quotient of groups,
then there is a natural quotient distance d'_p that metrizes the quotient
topology. On the other hand, the Finsler quotient metric defined in O provides
a way to measure curves, and therefore, there is an associated rectifiable
distance d_{O,p}. For , we prove that the distances d'_p and d_{O,p}
coincide. Based on this fact, we show that the metric space (O,d'_p) is a
complete path metric space. The other problem treated in this article is the
existence of metric geodesics, or curves of minimal length, in O. We give two
abstract partial results in this direction. The first concerns the initial
values problem and the second the fixed endpoints problem. We show how these
results apply to several examples. In the process, we improve some results
about the metric geometry of U with the p-norm.Comment: 30 pages. The examples in section 4 have been removed, those of
section 5 have been cut dow
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