Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$V = \oplus _{i\ge 1} V_i $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>=</mo> <msub> <mo>⊕</mo> <mrow> <mi>i</mi> <mo>≥</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>V</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be a graded vector space over a field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation> of characteristic 0, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ (\mathbb {L}(V), d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a differential free graded Lie algebra and (TV, d) its universal enveloping algebra. We define a multiplicative structure on the cochain complex <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ {{\,\textrm{Hom}\,}}_{TV}(TV \otimes (\mathbb {k}\oplus sV), TV)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mspace width="0.166667em" /> <mtext>Hom</mtext> <mspace width="0.166667em" /> </mrow> <mrow> <mi mathvariant="italic">TV</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mi>V</mi> <mo>⊗</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">k</mi> <mo>⊕</mo> <mi>s</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>T</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which yields the usual multiplication on the Hochschild cohomology <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$HH^(TV; TV)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <msup> <mi>H</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mi>V</mi> <mo>;</mo> <mi>T</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\mathbb {L}(V), d) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a Quillen model of a simply connected, compact and oriented manifold X, we recover the inclusion of the Lie algebra <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \pi _({{\,\textrm{aut}\,}}_1(X)) \otimes \mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>π</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <msub> <mrow> <mspace width="0.166667em" /> <mtext>aut</mtext> <mspace width="0.166667em" /> </mrow> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>⊗</mo> <mi mathvariant="double-struck">k</mi> </mrow> </math></EquationSource> </InlineEquation> in the free loop space homology <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ \mathbb {H}_*(LX, \mathbb {k})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi mathvariant="double-struck">H</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mi>X</mi> <mo>,</mo> <mi mathvariant="double-struck">k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Lie algebra cohomology and the product in the free loop space homology

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2025年10月9日

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Jean Baptiste Gatsinzi

摘要

Let $V = \oplus _{i\ge 1} V_i $ $V = \oplus_{i \geq 1} V_{i}$ be a graded vector space over a field $ \mathbb {k}$ $k$ of characteristic 0, $ (\mathbb {L}(V), d)$ $(L (V), d)$ be a differential free graded Lie algebra and (TV, d) its universal enveloping algebra. We define a multiplicative structure on the cochain complex $ {{\,\textrm{Hom}\,}}_{TV}(TV \otimes (\mathbb {k}\oplus sV), TV)$ ${Hom}_{TV} (T V \otimes (k \oplus s V), T V)$ which yields the usual multiplication on the Hochschild cohomology $HH^*(TV; TV)$ $H H^{*} (T V; T V)$ . Moreover if $(\mathbb {L}(V), d) $ $(L (V), d)$ is a Quillen model of a simply connected, compact and oriented manifold X, we recover the inclusion of the Lie algebra $ \pi _*({{\,\textrm{aut}\,}}_1(X)) \otimes \mathbb {k}$ $π_{*} ({aut}_{1} (X)) \otimes k$ in the free loop space homology $ \mathbb {H}_*(LX, \mathbb {k})$ $H_{*} (L X, k)$ .