79 research outputs found
Higman-Neumann-Neumann extension and embedding theorems for Leibniz algebras
In this work we introduce the Higman-Neumann-Neumann (HNN)-
extensions and the appropriate embedding theorems for dialgebras and
Leibniz algebras.
Due to the importance of the connection between the dialgebras and
Leibniz algebras and the relationship between associative algebras and
Lie algebras, we recall the theory of Groebner-Shirshov bases, and the
Composition-Diamond Lemma in associative algebras and Lie algebras,
as well as the theory of Groebner-Shirshov bases for dialgebras.
As an application of the HNN-extensions of dialgebras and Leibniz
algebras, we provide embedding theorems for dialgebras and Leibniz algebras,
respectively: every dialgebra embeds inside its any HNN-extension
and every Leibniz algebra embeds inside its any HNN-extension
Lie commutators in a free diassociative algebra
summary:We give a criterion for Leibniz elements in a free diassociative algebra. In the diassociative case one can consider two versions of Lie commutators. We give criterions for elements of diassociative algebras to be Lie under these commutators. One of them corresponds to Leibniz elements. It generalizes the Dynkin-Specht-Wever criterion for Lie elements in a free associative algebra
From Atiyah Classes to Homotopy Leibniz Algebras
A celebrated theorem of Kapranov states that the Atiyah class of the tangent
bundle of a complex manifold makes into a Lie algebra object in
, the bounded below derived category of coherent sheaves on .
Furthermore Kapranov proved that, for a K\"ahler manifold , the Dolbeault
resolution of is an
algebra. In this paper, we prove that Kapranov's theorem holds in much wider
generality for vector bundles over Lie pairs. Given a Lie pair , i.e. a
Lie algebroid together with a Lie subalgebroid , we define the Atiyah
class of an -module (relative to ) as the obstruction to
the existence of an -compatible -connection on . We prove that the
Atiyah classes and respectively make and
into a Lie algebra and a Lie algebra module in the bounded below
derived category , where is the abelian
category of left -modules and is the universal
enveloping algebra of . Moreover, we produce a homotopy Leibniz algebra and
a homotopy Leibniz module stemming from the Atiyah classes of and ,
and inducing the aforesaid Lie structures in .Comment: 36 page
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