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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:
17A30Nonassociative algebras satisfying other identities
55N35Other homology theories (algebraic topology)
55Q70Homotopy groups of special types
17A36Automorphisms, derivations, other operators
17B05Structure theory of Lie algebras