Unified products for Leibniz algebras. Applications

DOI10.1016/J.LAA.2013.07.021zbMATH Open1281.17003arXiv1307.2540OpenAlexW2018029201WikidataQ122958420 ScholiaQ122958420MaRDI QIDQ2435421

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Publication date: 19 February 2014

Published in: (Search for Journal in Brave)

Abstract: Let

m a t h f r a k g

be a Leibniz algebra and

E

a vector space containing

m a t h f r a k g

as a subspace. All Leibniz algebra structures on

E

containing

m a t h f r a k g

as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects:

m a t h c a l H {m a t h c a l L}_{m a t h f r a k g}^{2}, (V,, m a t h f r a k g)

provides the classification up to an isomorphism that stabilizes

m a t h f r a k g

and

m a t h c a l H {m a t h c a l L}^{2}, (V,, m a t h f r a k g)

will classify all such structures from the view point of the extension problem - here

V

is a complement of

m a t h f r a k g

in

E

. A general product, called the unified product, is introduced as a tool for our approach. The crossed (resp. bicrossed) products between two Leibniz algebras are introduced as special cases of the unified product: the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem. The description and the classification of all complements of a given extension

m a t h f r a k g s u b s e t e q m a t h f r a k E

of Leibniz algebras are given as a converse of the factorization problem. They are classified by another cohomological object denoted by

m a t h c a l H {m a t h c a l A}^{2} (m a t h f r a k h, m a t h f r a k g, |, (r i a n g l e r i g h t, r i a n g l e l e f t, l e f t h a r p o o n u p, i g h t h a r p o o n u p))

, where

(r i a n g l e r i g h t, r i a n g l e l e f t, l e f t h a r p o o n u p, i g h t h a r p o o n u p)

is the canonical matched pair associated to a given complement

m a t h f r a k h

. Several examples are worked out in details.

Full work available at URL: https://arxiv.org/abs/1307.2540

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