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<p>This paper develops a graphical calculus to determine the <i>n</i>-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley–Eilenberg algebra of an ordinary Lie algebra, we recover Safronov’s result that the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((n=1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>- and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((n=2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley–Eilenberg algebra of a Lie 2-algebra and obtain <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n\in \{1,2,3,4\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> shifted Poisson structures in this case, which we interpret as semi-classical data of ‘higher quantum groups’.</p>

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Shifted Poisson structures on higher Chevalley–Eilenberg algebras

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月18日

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  • Cameron Kemp,
  • Robert Laugwitz,
  • Alexander Schenkel

摘要

This paper develops a graphical calculus to determine the n-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley–Eilenberg algebra of an ordinary Lie algebra, we recover Safronov’s result that the \((n=1)\) ( n = 1 ) - and \((n=2)\) ( n = 2 ) -shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley–Eilenberg algebra of a Lie 2-algebra and obtain \(n\in \{1,2,3,4\}\) n ∈ { 1 , 2 , 3 , 4 } shifted Poisson structures in this case, which we interpret as semi-classical data of ‘higher quantum groups’.