Commutators of Pre-Lie n-Algebras and PL∞-Algebras

We show that a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></in...

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Main Authors:	Mengjun Wang, Zhixiang Wu
Format:	Article
Language:	English
Published:	MDPI AG 2025-05-01
Series:	Mathematics
Subjects:	<i>n</i>-ary algebras pre-Lie algebras left-symmetric algebras <i>L</i><sub>∞</sub>-algebras <i>A</i><sub>∞</sub>-algebras <i>PL</i><sub>∞</sub>-algebras
Online Access:	https://www.mdpi.com/2227-7390/13/11/1792
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author	Mengjun Wang Zhixiang Wu
author_facet	Mengjun Wang Zhixiang Wu
author_sort	Mengjun Wang
collection	DOAJ
description	We show that a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra <i>V</i> can be described by a nilpotent coderivation of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> on coalgebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>P</mi><mo>*</mo></msup><mi>V</mi></mrow></semantics></math></inline-formula>. Based on this result, we can generalise the result of Lada to show that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra carries a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra structure and every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra carries an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra structure. In particular, we obtain a pre-Lie <i>n</i>-algebra structure on an arbitrary partially associative <i>n</i>-algebra and deduce that pre-Lie <i>n</i>-algebras are <i>n</i>-Lie admissible.
format	Article
id	doaj-art-549be51da72c481fbd547652016deecf
institution	OA Journals
issn	2227-7390
language	English
publishDate	2025-05-01
publisher	MDPI AG
record_format	Article
series	Mathematics
spelling	doaj-art-549be51da72c481fbd547652016deecf2025-08-20T02:32:37ZengMDPI AGMathematics2227-73902025-05-011311179210.3390/math13111792Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-AlgebrasMengjun Wang0Zhixiang Wu1School of Mathematical Sciences, Nanjing University, Nanjing, 210008, ChinaSchool of Mathematical Sciences, Zhejiang University, Hangzhou 310027, ChinaWe show that a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra <i>V</i> can be described by a nilpotent coderivation of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> on coalgebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>P</mi><mo>*</mo></msup><mi>V</mi></mrow></semantics></math></inline-formula>. Based on this result, we can generalise the result of Lada to show that every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra carries a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra structure and every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>L</mi><mo>∞</mo></msub></mrow></semantics></math></inline-formula>-algebra carries an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mo>∞</mo></msub></semantics></math></inline-formula>-algebra structure. In particular, we obtain a pre-Lie <i>n</i>-algebra structure on an arbitrary partially associative <i>n</i>-algebra and deduce that pre-Lie <i>n</i>-algebras are <i>n</i>-Lie admissible.https://www.mdpi.com/2227-7390/13/11/1792<i>n</i>-ary algebraspre-Lie algebrasleft-symmetric algebras<i>L</i><sub>∞</sub>-algebras<i>A</i><sub>∞</sub>-algebras<i>PL</i><sub>∞</sub>-algebras
spellingShingle	Mengjun Wang Zhixiang Wu Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-Algebras Mathematics <i>n</i>-ary algebras pre-Lie algebras left-symmetric algebras <i>L</i><sub>∞</sub>-algebras <i>A</i><sub>∞</sub>-algebras <i>PL</i><sub>∞</sub>-algebras
title	Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-Algebras
title_full	Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-Algebras
title_fullStr	Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-Algebras
title_full_unstemmed	Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-Algebras
title_short	Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-Algebras
title_sort	commutators of pre lie i n i algebras and i pl i sub ∞ sub algebras
topic	<i>n</i>-ary algebras pre-Lie algebras left-symmetric algebras <i>L</i><sub>∞</sub>-algebras <i>A</i><sub>∞</sub>-algebras <i>PL</i><sub>∞</sub>-algebras
url	https://www.mdpi.com/2227-7390/13/11/1792
work_keys_str_mv	AT mengjunwang commutatorsofprelieinialgebrasandiplisubsubalgebras AT zhixiangwu commutatorsofprelieinialgebrasandiplisubsubalgebras

Commutators of Pre-Lie <i>n</i>-Algebras and <i>PL</i><sub>∞</sub>-Algebras

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