338 research outputs found
Left-symmetric algebroids
In this paper, we introduce a notion of a left-symmetric algebroid, which is
a generalization of a left-symmetric algebra from a vector space to a vector
bundle. The left multiplication gives rise to a representation of the
corresponding sub-adjacent Lie algebroid. We construct left-symmetric
algebroids from -operators on Lie algebroids. We study phase spaces
of Lie algebroids in terms of left-symmetric algebroids. Representations of
left-symmetric algebroids are studied in detail. At last, we study deformations
of left-symmetric algebroids, which could be controlled by the second
cohomology class in the deformation cohomology.Comment: 21 page
Rota-Baxter operators on Witt and Virasoro algebras
The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are
classified. As applications, the induced solutions of the classical Yang-Baxter
equation and the induced pre-Lie and PostLie algebra structures are obtained.Comment: 28 page
A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras
The approach for Poisson bialgebras characterized by Manin triples with
respect to the invariant bilinear forms on both the commutative associative
algebras and the Lie algebras is not available for giving a bialgebra theory
for transposed Poisson algebras. Alternatively, we consider Manin triples with
respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly,
we first introduce the notion of anti-pre-Lie bialgebras as the equivalent
structure of Manin triples of Lie algebras with respect to the commutative
2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras,
characterized by Manin triples of transposed Poisson algebras with respect to
the bilinear forms which are invariant on the commutative associative algebras
and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for
transposed Poisson algebras. Finally the coboundary cases and the related
structures such as analogues of the classical Yang-Baxter equation and
-operators are studied.Comment: 34 page
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