## On N-ary Algebras, Polyvector Gauge Theories in Noncommutative Clifford Spaces and Deformation Quantization

**Authors:** Carlos Castro

Polyvector-valued gauge field theories in noncommutative Clifford spaces
are presented. The noncommutative binary star products are associative
and require the use of the Baker-Campbell-Hausdorff formula. An
important relationship among the n-ary commutators of noncommuting
spacetime coordinates [X^{1},X^{2}, ......,X^{n}] and the poly-vector valued coordinates
X^{123...n} in noncommutative Clifford spaces is explicitly derived
and is given by [X^{1},X^{2}, ......,X^{n}] = n! X^{123...n}. It is argued how the
large N limit of n-ary commutators of n hyper-matrices X_{i1}_{i2}...._{in} leads
to Eguchi-Schild p-brane actions when p+1 = n. A noncomutative n-ary
generalized star product of functions is provided which is associated with
the deformation quantization of n-ary structures. Finally, brief comments
are made about the mapping of the Nambu-Heisenberg n-ary commutation
relations of linear operators into the deformed Nambu-Poisson
brackets of their corresponding symbols.

**Comments:** 13 Pages. This article has been submitted to Physics Letters B

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### Submission history

[v1] 11 Jan 2010

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