Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 023, 25 pages      arXiv:1111.6750      https://doi.org/10.3842/SIGMA.2012.023

Classification of Traces and Associated Determinants on Odd-Class Operators in Odd Dimensions

Carolina Neira Jiménez a and Marie Françoise Ouedraogo b
a) Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
b) Départment de Mathématiques, Université de Ouagadougou, 03 BP 7021, Burkina Faso

Received November 30, 2011, in final form April 11, 2012; Published online April 21, 2012

Abstract
To supplement the already known classification of traces on classical pseudodifferential operators, we present a classification of traces on the algebras of odd-class pseudodifferential operators of non-positive order acting on smooth functions on a closed odd-dimensional manifold. By means of the one to one correspondence between continuous traces on Lie algebras and determinants on the associated regular Lie groups, we give a classification of determinants on the group associated to the algebra of odd-class pseudodifferential operators with fixed non-positive order. At the end we discuss two possible ways to extend the definition of a determinant outside a neighborhood of the identity on the Lie group associated to the algebra of odd-class pseudodifferential operators of order zero.

Key words: pseudodifferential operators; odd-class; trace; determinant; logarithm; regular Lie group.

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