Noncommutative cohomology and electromagnetism on $C_q[SL_2]$ at roots of unity
Abstract
We compute the noncommutative de Rham cohomology for the finitedimensional qdeformed coordinate ring $C_q[SL_2]$ at odd roots of unity and with its standard 4dimensional differential structure. We find that $H^1$ and $H^3$ have three additional modes beyond the generic $q$case where they are 1dimensional, while $H^2$ has six additional modes. We solve the spin0 and Maxwell theory on $C_q[SL_2]$ including a complete picture of the selfdual and antiself dual solutions and of Lorentz and temporal gauge fixing. The system behaves in fact like a noncompact space with selfpropagating modes (i.e., in the absence of sources). We also solve with examples of `electric' and `magnetic' sources including the biinvariant element $\theta\in H^1$ which we find can be viewed as a source in the local (Minkowski) timedirection (i.e. a uniform electric charge density).
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2001
 arXiv:
 arXiv:math/0110323
 Bibcode:
 2001math.....10323G
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Differential Geometry;
 High Energy Physics  Theory
 EPrint:
 19 pages LaTeX, no figures