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On the Novikov algebra structures adapted to the automorphism structure of a Lie group. (English) Zbl 1033.17002
Summary: Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra in which there exists a special affine structure (connection with zero curvature and torsion) defined by the Novikov algebra. For ensuring the consequences for the group structure, we need to consider the more intrinsic connections defined by Novikov algebra structures, that is, the connections which are adapted to the automorphism structure of a Lie group. The resulting Novikov algebra is called a derivation algebra every left multiplication operator of which is a derivation of its sub-adjacent Lie algebra. In this paper, we commence a study of the Novikov derivation algebras and, as a consequence, construct Novikov algebras on some 2-solvable Lie algebras.
##### MSC:
 17A30 Nonassociative algebras satisfying other identities 55N35 Other homology theories (algebraic topology) 55Q70 Homotopy groups of special types 17A36 Automorphisms, derivations, other operators 17B05 Structure theory of Lie algebras
##### Keywords:
Novikov algebras; Novikov derivation algebras