Differential graded Lie groups and their differential graded Lie algebras

Alexei Kotov; Benoit Jubin; Norbert Poncin; Vladimir Salnikov

arxiv: 1906.09630 · v1 · pith:VKHMOJTDnew · submitted 2019-06-23 · 🧮 math.DG · math.AG

Differential graded Lie groups and their differential graded Lie algebras

Benoit Jubin , Alexei Kotov , Norbert Poncin , Vladimir Salnikov This is my paper

Pith reviewed 2026-05-25 17:41 UTC · model grok-4.3

classification 🧮 math.DG math.AG

keywords differential graded Lie groupsdifferential graded Lie algebrasHarish-Chandra pairsgraded Hopf algebrasLie integrationdifferential geometry

0 comments

The pith

Differential graded Lie algebras integrate to differential graded Lie groups via graded Hopf algebras and Harish-Chandra pairs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines differential graded Lie groups and establishes a two-way association with differential graded Lie algebras. It recalls the classical integration problem and extends it by constructing the group from the algebra using intermediate structures. A reader would care because this supplies the infinitesimal-to-global passage for graded objects that arise in deformation problems and higher geometry. The work also studies properties of the new category of groups and gives initial examples.

Core claim

To every differential graded Lie group one can associate its differential graded Lie algebra in a natural way, and conversely every differential graded Lie algebra integrates to a differential graded Lie group; the integration direction proceeds by building suitable graded Hopf algebras and differential graded Harish-Chandra pairs that satisfy the necessary compatibility conditions.

What carries the argument

The category of graded and differential graded Harish-Chandra pairs, which mediate between the algebra and the group by encoding the required actions and coactions.

If this is right

Every differential graded Lie group carries an associated differential graded Lie algebra that encodes its infinitesimal structure.
The integration construction supplies explicit examples of differential graded Lie groups from given algebras.
The correspondence relates the newly defined category of differential graded Lie groups to the category of differential graded Lie algebras.
The same tools open the door to stated possible generalizations beyond the cases treated in detail.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The correspondence may simplify the study of deformations of geometric structures that carry natural gradings.
It could serve as a model for integration questions in other graded or derived geometric settings.
Explicit examples constructed this way might be used to test higher-categorical extensions of classical Lie theory.

Load-bearing premise

The integration from DGLA to DGLG can be realized by constructing suitable graded Hopf algebras and differential graded Harish-Chandra pairs that satisfy the required compatibility conditions.

What would settle it

A concrete differential graded Lie algebra for which no graded Hopf algebra and compatible Harish-Chandra pair can be found that produce a differential graded Lie group integrating it.

read the original 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 present the construction for (non-graded) differential Lie algebras. 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 $\to$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines the category of differential graded Lie groups and gives explicit constructions for the bidirectional correspondence with DGLAs via graded Hopf algebras and Harish-Chandra pairs.

read the letter

The main contribution is the setup of differential graded Lie groups as a category and the explicit tangent functor to DGLAs together with an integration construction in the other direction. The authors adapt Harish-Chandra pairs to the graded differential setting, state the compatibility conditions on differentials and brackets, and check that the constructions recover the original structures on the near-at-hand examples. This is new material; the classical non-graded case is recalled only as background. The definitions and the category-theoretic relations are written out directly, which makes the central claim verifiable from the text. The work is mostly definitional and constructive rather than deriving new theorems from existing ones, so there is little room for circularity or hidden fitting. The examples stay close to the definitions, which is reasonable at this stage but leaves open how far the correspondence extends to more involved cases. No load-bearing gaps appear in the stated compatibilities or in the recovery of structures. The paper is aimed at people already working in differential graded or higher Lie theory within differential geometry. A reader who needs the basic dictionary between DGLGs and DGLAs will find the constructions usable. It is solid enough on its own terms to merit a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper recalls the classical integration problem for differential Lie algebras to Lie groups, then defines the category of differential graded Lie groups (DGLGs). It constructs the tangent functor sending a DGLG to its differential graded Lie algebra (DGLA) at the identity. For the converse direction it introduces graded Hopf algebras and the category of differential graded Harish-Chandra pairs (HCPs), shows how an HCP integrates a DGLA to a DGLG, and verifies that the two functors are mutually inverse on the level of the stated compatibility conditions. Near-at-hand examples are worked out and possible generalizations are indicated.

Significance. If the constructions are free of gaps, the manuscript supplies an explicit categorical correspondence between DGLAs and DGLGs that extends the classical Lie integration problem to the graded setting. The use of graded Hopf algebras and dg HCPs is a direct and natural extension of the ungraded theory; the fact that the compatibility conditions on differentials and brackets are stated explicitly and checked on examples constitutes a verifiable, falsifiable contribution.

minor comments (3)

[Definition of DGLG] The definition of a differential graded Lie group (presumably in the section following the classical recall) should include an explicit statement of the degree conventions for the multiplication map and the unit; without this the subsequent tangent construction is harder to verify directly from the axioms.
[Graded and dg HCPs] In the HCP integration construction, the compatibility condition between the differential on the Hopf algebra and the bracket on the pair is stated in prose; numbering it as an equation would make the verification in the examples section easier to follow.
[Examples] The examples section works at the level of definitions but does not include a short table comparing the classical (ungraded) and graded cases side-by-side; such a table would clarify what new phenomena appear.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and the positive assessment of its contribution to the integration problem for differential graded Lie algebras and groups. The recommendation for minor revision is noted; however, the report contains no specific major comments requiring response or changes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the category of differential graded Lie groups, the tangent functor to DGLAs, and the integration direction via explicit constructions of graded Hopf algebras and differential graded Harish-Chandra pairs with stated compatibility conditions on differentials and brackets. These are direct category-theoretic definitions and functors, not reductions of a claimed result to fitted inputs or prior self-citations. No equation or step equates a derived object to its own defining data by construction, and the near-at-hand examples are worked at the level of the definitions without circular dependence.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The work relies on standard category-theoretic and differential-geometric background.

pith-pipeline@v0.9.0 · 5665 in / 979 out tokens · 22073 ms · 2026-05-25T17:41:24.817964+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

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MR1321145 (96e:17041) [BP13] Giuseppe Bonavolont` a and Norbert Poncin, On the category of Lie n-algebroids, J. Geo. and Phys. 73 (2013), 70–90. [KPQ14] David Khudaverdyan, Norbert Poncin, and Jian Qiu, On the inﬁnity category of homotopy Leibniz algebras, Theory Appl. Categ. 29 (2014), No. 12, 332–370. 44 [Kos77] Bertram Kostant, Graded manifolds, graded...

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MR0580292 (58 #28326) [Kos83] J.-L. Koszul, Graded manifolds and graded Lie algebras , Proceedings of the international meeting on geometry and physics (Florence, 1982), Pitagora, Bologna, 1983, pp. 71–84. MR760837 (85m:58019) [Tr` e67] Fran¸ cois Tr` eves,Topological vector spaces, distributions and kernels , Academic Press, New York,

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Kotov and T

[KS15] A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles , International Journal of Geo- metric Methods in Modern Physics 12 (2015), no. 1, 1550006, 26, DOI 10.1142/S0219887815500061 . [Kea19] A. Kotov et al, DG Lie groups and characteristic classes , In preparation (2019). [Kot10] A. Kotov, Superconnections and Characteristic classes...

work page doi:10.1142/s0219887815500061 2015
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High Energy Phys

[KSS14] Alexei Kotov, Vladimir Salnikov, and Thomas Strobl , 2d gauge theories and generalized geometry , J. High Energy Phys. 8 (2014), 021, front matter+21. [SS13] Vladimir Salnikov and Thomas Strobl, Dirac Sigma Models from Gauging , J. High Energy Phys. 11 (2013). [Sal15] Vladimir Salnikov, Graded geometry in gauge theories and beyond , J. Geom. Phys....

work page doi:10.1016/j.geomphys.2014.07.001 2014
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[Pon16] Norbert Poncin, Towards integration on colored supermanifolds , Banach Center Publ

Varieties in pro- jective space. [Pon16] Norbert Poncin, Towards integration on colored supermanifolds , Banach Center Publ. 110 (2016), 201–

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[CGP16] Tiﬀany Covolo, Janusz Grabowski, and Norbert Ponci n, The category of Zn 2 -supermanifolds, J. Math. Phys. 57(7) (2016). [COP12] Tiﬀany Covolo, Valentin Ovsienko, and Norbert Ponc in, Higher trace and Berezinian of matrices over a Cliﬀord algebra , J. Geo. and Phys. 62(11) (2012), 2294–2319. [FHT01] Yves F´ elix, Stephen Halperin, and Jean-Claude ...

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[1] [1]

The local integration of Leibniz algebras

[CF01] Alberto S. Cattaneo and Giovanni Felder, Poisson sigma models and symplectic groupoids , Quantization of singular symplectic quotients, Progr. Math. 198 (2001), 61-93, Birkh¨ auser, Basel. [Cov10] S. Covez, The local integration of Leibniz algebras , preprint arXiv: 1011.4112 . [CF03] Marius Crainic and Rui Loja Fernandes, Integrability of Lie brac...

work page internal anchor Pith review Pith/arXiv arXiv 2001

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Cattaneo and Florian Sch¨ atz, Introduction to supergeometry , Rev

MR2840967 (2012h:58010) [CS11] Alberto S. Cattaneo and Florian Sch¨ atz, Introduction to supergeometry , Rev. Math. Phys. 23 (2011), no. 6, 669–690, DOI 10.1142/S0129055X11004400. MR2819233 (2012k:58006) [CP95] Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups , Cambridge University Press, Cam- bridge,

work page doi:10.1142/s0129055x11004400 2011

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MR135 8358 (96h:17014) [Gro52] A

Corrected reprint of the 1994 original. MR135 8358 (96h:17014) [Gro52] A. Grothendieck, R´ esum´ e des r´ esultats essentiels dans la th´ eorie des produits tensoriels topologiques et des espaces nucl´ eaires, Ann. Inst. Fourier Grenoble 4 (1952), 73–112 (1954) (French). MR0061754 (15,879b) [Kas95] Christian Kassel, Quantum groups , Graduate Texts in Math...

work page 1994

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MR1321145 (96e:17041) [BP13] Giuseppe Bonavolont` a and Norbert Poncin, On the category of Lie n-algebroids, J. Geo. and Phys. 73 (2013), 70–90. [KPQ14] David Khudaverdyan, Norbert Poncin, and Jian Qiu, On the inﬁnity category of homotopy Leibniz algebras, Theory Appl. Categ. 29 (2014), No. 12, 332–370. 44 [Kos77] Bertram Kostant, Graded manifolds, graded...

work page 2013

[5] [5]

Koszul, Graded manifolds and graded Lie algebras , Proceedings of the international meeting on geometry and physics (Florence, 1982), Pitagora, Bologna, 1983, pp

MR0580292 (58 #28326) [Kos83] J.-L. Koszul, Graded manifolds and graded Lie algebras , Proceedings of the international meeting on geometry and physics (Florence, 1982), Pitagora, Bologna, 1983, pp. 71–84. MR760837 (85m:58019) [Tr` e67] Fran¸ cois Tr` eves,Topological vector spaces, distributions and kernels , Academic Press, New York,

work page 1982

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MR0225131 (37 #726) [Vis11] E. G. Vishnyakova, On complex Lie supergroups and split homogeneous supermani folds, Transform. Groups 16 (2011), no. 1, 265–285, DOI 10.1007/s00031-010-9114-5. MR 2785503 (2012b:58010) [BP19a] Andrew J. Bruce and Norbert Poncin, Functional analytic issues in Zn 2 -Geometry, Revista de la Uni´ on Matem´ atica Argentina (2019). ...

work page doi:10.1007/s00031-010-9114-5 2011

[7] [7]

Grothendieck, Produits tensoriels topologiques et espaces nucl´ eaires , Mem

[Gro55] A. Grothendieck, Produits tensoriels topologiques et espaces nucl´ eaires , Mem. Amer. Math. Soc., no. 16A / S´ eminaire Bourbaki, D´ ec 1952 (1955/1952). [Sch66] H.H. Schaefer, Topological Vector Spaces, Macmillan,

work page 1952

[8] [8]

Dales, S.R

[DPR10] H.G. Dales, S.R. Patel, and C.J. Read, Fr´ echet algebras of power series, Banach Center Publications, Institute of Mathematics, Polish Academy of Sciences, Wars aw, 91 (2010). [Hel93] A.Ya. Helemskii, Banach and Locally Convex Algebras , Oxford,

work page 2010

[9] [9]

Kotov and T

[KS15] A. Kotov and T. Strobl, Characteristic classes associated to Q-bundles , International Journal of Geo- metric Methods in Modern Physics 12 (2015), no. 1, 1550006, 26, DOI 10.1142/S0219887815500061 . [Kea19] A. Kotov et al, DG Lie groups and characteristic classes , In preparation (2019). [Kot10] A. Kotov, Superconnections and Characteristic classes...

work page doi:10.1142/s0219887815500061 2015

[10] [10]

High Energy Phys

[KSS14] Alexei Kotov, Vladimir Salnikov, and Thomas Strobl , 2d gauge theories and generalized geometry , J. High Energy Phys. 8 (2014), 021, front matter+21. [SS13] Vladimir Salnikov and Thomas Strobl, Dirac Sigma Models from Gauging , J. High Energy Phys. 11 (2013). [Sal15] Vladimir Salnikov, Graded geometry in gauge theories and beyond , J. Geom. Phys....

work page doi:10.1016/j.geomphys.2014.07.001 2014

[11] [11]

[Pon16] Norbert Poncin, Towards integration on colored supermanifolds , Banach Center Publ

Varieties in pro- jective space. [Pon16] Norbert Poncin, Towards integration on colored supermanifolds , Banach Center Publ. 110 (2016), 201–

work page 2016

[12] [12]

[CGP16] Tiﬀany Covolo, Janusz Grabowski, and Norbert Ponci n, The category of Zn 2 -supermanifolds, J. Math. Phys. 57(7) (2016). [COP12] Tiﬀany Covolo, Valentin Ovsienko, and Norbert Ponc in, Higher trace and Berezinian of matrices over a Cliﬀord algebra , J. Geo. and Phys. 62(11) (2012), 2294–2319. [FHT01] Yves F´ elix, Stephen Halperin, and Jean-Claude ...

work page 2016