<p>It is shown that any Lie affgebra, that is an algebraic system consisting of an affine space together with a bi-affine multiplication satisfying affine versions of the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together with an element and a specific generalized derivation (in the sense of Leger and Luks in J Algebra 228:165–203, 2000). These Lie algebraic data can be taken for the construction of a Lie affgebra or, conversely, they can be uniquely derived for any Lie algebra fibre of the Lie affgebra. The close relationship between Lie affgebras and (enriched by the additional data) Lie algebras can be employed to attempt a classification of the former by the latter. In particular, up to isomorphism, a complex Lie affgebra with a simple Lie algebra fibre <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2377_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> is fully determined by a scalar and an element of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2377_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> fixed up to an automorphism of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2377_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation>, and it can be universally embedded in a trivial extension of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2377_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">g</mi> </math></EquationSource> </InlineEquation> by a derivation. The study is illustrated by a number of examples that include all Lie affgebras with one-dimensional, non-abelian two-dimensional, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2377_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {s}\mathfrak {l}(2,\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2377_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {s}\mathfrak {o}(3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> Lie algebra fibres. Extensions of Lie affgebras by cocycles and their relation to cocycle extensions of tangent Lie algebras is briefly discussed and illustrated by Lie affgebras with the Witt and Virasoro algebra fibres.</p>

Lie Affgebras Vis-à-Vis Lie Algebras

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2025年2月28日

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Ryszard R. Andruszkiewicz,
Tomasz Brzeziński,
Krzysztof Radziszewski

摘要

It is shown that any Lie affgebra, that is an algebraic system consisting of an affine space together with a bi-affine multiplication satisfying affine versions of the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together with an element and a specific generalized derivation (in the sense of Leger and Luks in J Algebra 228:165–203, 2000). These Lie algebraic data can be taken for the construction of a Lie affgebra or, conversely, they can be uniquely derived for any Lie algebra fibre of the Lie affgebra. The close relationship between Lie affgebras and (enriched by the additional data) Lie algebras can be employed to attempt a classification of the former by the latter. In particular, up to isomorphism, a complex Lie affgebra with a simple Lie algebra fibre $\mathfrak {g}$ $g$ is fully determined by a scalar and an element of $\mathfrak {g}$ $g$ fixed up to an automorphism of $\mathfrak {g}$ $g$ , and it can be universally embedded in a trivial extension of $\mathfrak {g}$ $g$ by a derivation. The study is illustrated by a number of examples that include all Lie affgebras with one-dimensional, non-abelian two-dimensional, $\mathfrak {s}\mathfrak {l}(2,\mathbb {C})$ $s l (2, C)$ and $\mathfrak {s}\mathfrak {o}(3)$ $s o (3)$ Lie algebra fibres. Extensions of Lie affgebras by cocycles and their relation to cocycle extensions of tangent Lie algebras is briefly discussed and illustrated by Lie affgebras with the Witt and Virasoro algebra fibres.