Multilinear quantum Lie operations

The notion of multilinear quantum Lie operation appears naturally in connection with a different attempts to generalize the Lie algebras. There are a number of reasons why the generalizations are necessary. First of all this is the demands for a ”quantum algebra” which was formed in the papers by Ju.I. Manin, V.G. Drinfeld, S.L. Woronowicz, G. Lusztig, L.D. Faddeev, and many others. These demands are defined by a desire to keep the intuition of the quantum mechanics differential calculus that is based on the fundamental concepts of the Lie groups and Lie algebras theory. Normally the quest for definition of bilinear brackets on the module of differential 1-forms that replace the Lie operation leads to restrictions like multiplicative skew-symmetry of the quantization parameters [1], involutivity of braidings, or bicovariancy of the differential calculus [2], [3]. At the same time lots of quantizations, for example the Drinfeld–Jimbo one, are defined by multiplicative non skew-symmetric parameters, and they define not bicovariant (but one-sided covariant) calculus. By these, and of course, by many other reasons, the attention of researches has been extended on operations that replace the Lie brackets, but depend on greater number of variables. Such are, for example, n-Lie algebras introduced by V.T. Filippov [4]