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DIFFERENTIAL GRADED LIE GROUPS AND THEIR DIFFERENTIAL GRADED LIE ALGEBRAS

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Abstract

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and recollect the known results. Then, we define the category of differential graded Lie groups and study its properties. We show how to associate a differential graded Lie algebra to every differential graded Lie group and vice versa. For the DGLA → DGLG direction, the main “tools” are graded Hopf algebras and Harish-Chandra pairs (HCP)—we define the category of graded and differential graded HCPs and explain how those are related to the desired construction. We describe some near-at-hand examples and mention possible generalizations.

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Authors and Affiliations

  1. Institut de Mathématiques, de Jussieu—Paris Rive Gauche, 4 place Jussieu, B.C. 247, 75252, Paris Cedex 5, France

    BENOIT JUBIN

  2. Faculty of Science, University of Hradec Kralove, Rokitanskeho 62, 50003, Hradec Kralove, Czech Republic

    ALEXEI KOTOV

  3. RMATH, FSTC, Université du Luxembourg, Maison du Nombre, 6, Avenue de la Fonte, L-4364, Esch-sur-Alzette, Luxembourg

    NORBERT PONCIN

  4. LaSIE—CNRS & La Rochelle University, Av. Michel Crépeau, 17042, La Rochelle Cedex 1, France

    VLADIMIR SALNIKOV

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  1. BENOIT JUBIN
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  2. ALEXEI KOTOV
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  3. NORBERT PONCIN
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  4. VLADIMIR SALNIKOV
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Correspondence to VLADIMIR SALNIKOV.

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JUBIN, B., KOTOV, A., PONCIN, N. et al. DIFFERENTIAL GRADED LIE GROUPS AND THEIR DIFFERENTIAL GRADED LIE ALGEBRAS. Transformation Groups 27, 497–523 (2022). https://doi.org/10.1007/s00031-021-09666-9

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