367 research outputs found
Delzant's T-invariant, Kolmogorov complexity and one-relator groups
We prove that ``almost generically'' for a one-relator group Delzant's
-invariant (which measures the smallest size of a finite presentation for a
group) is comparable in magnitude with the length of the defining relator. The
proof relies on our previous results regarding isomorphism rigidity of generic
one-relator groups and on the methods of the theory of Kolmogorov-Chaitin
complexity. We also give a precise asymptotic estimate (when is fixed and
goes to infinity) for the number of isomorphism classes of
-generator one-relator groups with a cyclically reduced defining relator of
length : Here
means that .Comment: A revised version, to appear in Comment. Math. Hel
Cartan Calculus on Quantum Lie Algebras
A generalization of the differential geometry of forms and vector fields to
the case of quantum Lie algebras is given. In an abstract formulation that
incorporates many existing examples of differential geometry on quantum spaces
we combine an exterior derivative, inner derivations, Lie derivatives, forms
and functions all into one big algebra, the ``Cartan Calculus''. (This is an
extended version of a talk presented by P. Schupp at the XXII
International Conference on Differential Geometric Methods in Theoretical
Physics, Ixtapa, Mexico, September 1993)Comment: 15 pages in LaTeX, LBL-34833 and UCB-PTH-93/3
Cartan Calculus for Hopf Algebras and Quantum Groups
A generalization of the differential geometry of forms and vector fields to
the case of quantum Lie algebras is given. In an abstract formulation that
incorporates many existing examples of differential geometry on quantum groups,
we combine an exterior derivative, inner derivations, Lie derivatives, forms
and functions all into one big algebra. In particular we find a generalized
Cartan identity that holds on the whole quantum universal enveloping algebra of
the left-invariant vector fields and implicit commutation relations for a
left-invariant basis of 1-forms.Comment: 15 pages (submitted to Comm. Math. Phys.
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