Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

Shahn Majid

Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

Quantum Algebra 2007-05-23 v2 High Energy Physics - Theory Differential Geometry

Authors: Shahn Majid

Abstract

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$ , using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for $q$ an odd $r$ 'th root of unity that its eigenvalues are given by $q$ -integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant background curvature. We fully solve the Dirac equation for $r=3$ .

Keywords

noncommutative Ricci curvature Dirac operator quantum group C_q[SL_2] Killing metric roots of unity Dirac equation

Cite

@article{arxiv.math/0206187,
  title  = {Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity},
  author = {Shahn Majid},
  journal= {arXiv preprint arXiv:math/0206187},
  year   = {2007}
}

Comments

16 pages amslatex. Minor revision to Dirac operator, now solving fully for $r=3$

Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

Abstract

Keywords

Cite

Comments

Related papers