×

A higher order analog of the Chevalley-Eilenberg complex and the deformation theory of \(n\)-gebras. (English) Zbl 0833.17021

St. Petersbg. Math. J. 6, No. 2, 429-438 (1995); translation from Algebra Anal. 6, No. 2, 262-272 (1994).
It is shown how basic homological constructions for Lie algebras are generalized to the case of Nambu-Lie gebras. To simplify the presentation, the author confines himself to the case \(n=3\); all the results proved can be generalized to the higher order case. It is pointed out that the obtained results are far from being complete and should be considered only as a first step towards developing a (co)homology theory for higher order algebraic operations.
In section 2 he introduces an analog of Chevalley-Eilenberg complex (with trivial coefficients) for Nambu-Lie gebras [cf. the author, Commun. Math. Phys. 160, 295-315 (1993; Zbl 0808.70015)]; the proof of the boundary property is simple and follows classical arguments. In section 3 he presents results on the analog of Hochschild cohomology for higher order algebraic operations. Starting with the property that the skew-symmetric part of a Hochschild 2-cocycle (with coefficients \({\mathcal A}\)) is a biderivation, he defines analogs of associative algebras and Hochschild cohomology with the property that the totally skew-symmetric part of a 2- cocycle is a three-derivation. Another important ingredient of the deformation theory is provided by the fact that the skew-symmetric part of the Hochschild 2-cocycle arising from a \(k\)-algebra deformation satisfies the Jacobi identity. Finally he discusses a possible generalization of Weyl quantization for the case of higher order algebraic operations.

MSC:

17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
70H99 Hamiltonian and Lagrangian mechanics
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 0808.70015
PDFBibTeX XMLCite