Noncommutative cohomology and electromagnetism on $C_q[SL_2]$ at roots of unity

Details Noncommutative cohomology and electromagnetism on $C_q[SL_2]$ at roots of unity Noncommutative cohomology and electromagnetism on $C_q[SL_2]$ at roots of unity

2001-10-31

arxiv.org/abs/math/0110323v1

Lett.Math.Phys. 60 (2002) 221-237

Mathematics

Quantum Algebra

19 pages LaTeX, no figures

Scientific paper

We compute the noncommutative de Rham cohomology for the finite-dimensional q-deformed coordinate ring $C_q[SL_2]$ at odd roots of unity and with its standard 4-dimensional differential structure. We find that $H^1$ and $H^3$ have three additional modes beyond the generic $q$-case where they are 1-dimensional, while $H^2$ has six additional modes. We solve the spin-0 and Maxwell theory on $C_q[SL_2]$ including a complete picture of the self-dual and anti-self dual solutions and of Lorentz and temporal gauge fixing. The system behaves in fact like a noncompact space with self-propagating 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) time-direction (i.e. a uniform electric charge density).

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