Poincare-Birkhoff-Witt theorem for pre-Lie and postLie algebras

Gubarev, Vsevolod

doi:10.48550/arxiv.1903.02960

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Introduction1

Derived Algebras and Manin White Product Of Binary Operads1

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2024

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Cited by 1 publication

(2 citation statements)

References 30 publications

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“…For example, every preAs-algebra with respect to the "dendriform commutator" a ⊢ b − b ⊣ a is a preLie-algebra. The properties of the corresponding left adjoint functor (universal envelope) are close to what we have for ordinary Lie algebras [6,12]. The class of pre-Novikov algebras has been recently studied in [24]: it coincides with DZinb.…”

Section: Introductionsupporting

confidence: 61%

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On Pre-Novikov Algebras and Derived Zinbiel Variety

Kolesnikov,

Mashurov,

Sartayev

2024

SIGMA

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For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation.

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Section: Introductionsupporting

confidence: 61%

“…2,2 ν (12) , µ, ν = γ 4 2,2 (x 2 * x 1 , x 1 x 2 , x 1 * x 2 ) = (x 3 * x 4 ) * (x 1 x 2 ), and so on.…”

Section: Derived Algebras and Manin White Product Of Binary Operadsmentioning

confidence: 99%