On M-Algebras, the Quantisation of Nambu-Mechanics, and Volume Preserving Diffeomorphisms

Abstract

M-branes are related to theories on function spaces $\mathcal{A}$ involving M-linear non-commutative maps from $\mathcal{A}\times \cdots \times \mathcal{A}$ to $\mathcal{A}$. While the Lie-symmetry-algebra of volume preserving diffeomorphisms of $T^M$ cannot be deformed when M>2, the arising M-algebras naturally relate to Nambu's generalisation of Hamiltonian mechanics, e.g. by providing a representation of the canonical M-commutation relations, $[J_1,\cdots ,J_M]=i\hslash$. Concerning multidimensional integrability, an important generalisation of Lax-pairs is given.

16 pages, LaTex