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Atiyah classes of strongly homotopy Lie pairs

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Zhuo Chen, Honglei Lang, Maosong Xiang

Quantum Algebra · Submitted 4 Sep 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the Atiyah class $[\alpha^{L/A}]$ induces a graded Lie algebra structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and the Atiyah class $[\alpha^E]$ of any $A$-module $E$ induces a Lie algebra module structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,E)$.

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arXiv ID1609.00984 (version 4)
Pith IDWVRWUJGN, Pith's permanent record of this paper and its review
CitationHonglei Lang, Maosong Xiang, Zhuo Chen. "Atiyah classes of strongly homotopy Lie pairs." arXiv:1609.00984 (2016). https://pith.science/paper/WVRWUJGN
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,538 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.75 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. In fact, given such an SH Lie pair $(L, A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[\alpha^E]$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class $[\alpha^{L/A}]$ induces a graded Lie algebra structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and the Atiyah class $[\alpha^E]$ of any $A$-module $E$ induces a Lie algebra module structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,E)$. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. background 1.00 0.06 0.02 asserted 0.99
s1 We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. contribution 0.98 0.94 0.20 asserted 1.00
s2 In fact, given such an SH Lie pair $(L, A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[\alpha^E]$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. contribution 0.62 0.81 0.19 asserted 1.00
s3 We show that the Atiyah class $[\alpha^{L/A}]$ induces a graded Lie algebra structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and the Atiyah class $[\alpha^E]$ of any $A$-module $E$ induces a Lie algebra module structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,E)$. result 0.99 0.95 0.51 asserted 1.00
s4 Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$. result 1.00 0.87 0.13 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

The subject of this paper is strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra $A$ when it is extended to $L$. In fact, given such an SH Lie pair $(L, A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[\alpha^E]$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. We show that the Atiyah class $[\alpha^{L/A}]$ induces a graded Lie algebra structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,L/A[-2])$, and the Atiyah class $[\alpha^E]$ of any $A$-module $E$ induces a Lie algebra module structure on $\operatorname{H}^\bullet_{\mathrm{CE}}(A,E)$. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.

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