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

Abstract

M-branes are related to theories on function spaces $\cal{A}$ involving M-linear non-commutative maps from $\cal{A} \times \cdots \times \cal{A}$ to $\cal{A}$. 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, $[J_1,\cdots, J_M]=i\hbar$. Concerning multidimensional integrability, an important generalisation of Lax-pairs is given.