The Dirac Operator on SUq(2)

2004/11/30 by Ludwik Dabrowski, Ludwik Dąbrowski, Giovanni Landi +4
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Biology #Computer science #Dirac operator #Mathematical physics #Mathematics #Operator (biology) #Physics #Spectral Theory in Mathematical Physics #math.QA #msc:17B37 #msc:58B34

paper · pdf · doi:10.1007/s00220-005-1383-9

published as Commun.Math.Phys. 259 (2005) 729-759 · v2: minor changes; to appear in CMP

arxiv created 2005/02/03 · openalex publication_date 2005/06/20 · arxiv updated 2009/12/01 · openalex created_date 2021/02/01 · openalex updated_date 2026/08/05

Abstract

We construct a 3⁺ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

Citations

• Some Noncommutative Geometric Aspects of SU_q(2)
• Tomita's Theory of Modular Hilbert Algebras and its Applications
• Quantum Groups