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A Lie algebraic approach to Novikov algebras. (English) Zbl 1033.17001
Summary: Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra. Thus it is useful to relate the study of Novikov algebras to the theory of Lie algebras. In this paper, we realize Novikov algebras through a Lie algebraic approach. Such a realization could be important in physics and geometry. We find that all transitive Novikov algebras in dimension 3 can be realized as the Novikov algebras obtained through Lie algebras and their compatible linear (global) deformations.
MSC:
17A30Nonassociative algebras satisfying other identities
55N35Other homology theories (algebraic topology)
55Q70Homotopy groups of special types