June  2022, 14(2): 273-305. doi: 10.3934/jgm.2022008

Modular class of Lie $ \infty $-algebroids and adjoint representations

1. 

University of Coimbra, CMUC, Department of Mathematics, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal

2. 

Institut Élie Cartan de Lorraine, UMR 7502 du CNRS, Université de Lorraine, Metz

Received  August 2021 Published  June 2022 Early access  April 2022

Fund Project: Raquel Caseiro is partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Camille Laurent-Gengoux thanks the "Granum" project MITI 80 primes CNRS for its financial support while this project was completed

We study the modular class of $ Q $-manifolds, and in particular of negatively graded Lie $ \infty $-algebroid. We show the equivalence of several descriptions of those classes, that it matches the classes introduced by various authors and that the notion is homotopy invariant. In the process, the adjoint and coadjoint actions up to homotopy of a Lie $ \infty $-algebroid are spelled out. We also wrote down explicitly some dualities, e.g. between representations up to homotopies of Lie $ \infty $-algebroids and their $ Q $-manifold equivalent, which we hope to be of use for future reference.

Citation: Raquel Caseiro, Camille Laurent-Gengoux. Modular class of Lie $ \infty $-algebroids and adjoint representations. Journal of Geometric Mechanics, 2022, 14 (2) : 273-305. doi: 10.3934/jgm.2022008
References:
[1]

C. Abad and M. Crainic, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math., 663 (2012), 91-126.  doi: 10.1515/CRELLE.2011.095.

[2]

I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math., 626 (2009), 1-37.  doi: 10.1515/CRELLE.2009.001.

[3]

I. Androulidakis and M. Zambon, Holonomy transformations for singular foliations, Advances in Mathematics, 256 (2014), 348-397.  doi: 10.1016/j.aim.2014.02.003.

[4]

M.-J. Azimi, C. Laurent-Gengoux and J.-M.Nunes da Costa, Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids, Communications in Algebra, 46 (2018), issue 2,494–515. doi: 10.1080/00927872.2017.1288234.

[5]

G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids, J. Geom. Phys., 73 (2013), 70-90.  doi: 10.1016/j.geomphys.2013.05.004.

[6]

J. Block, Duality and equivalence of module categories in noncommutative geometry, Memorial volume for R. Bott., CRM Proc. Lecture Notes, 50, Amer. Math. Soc., Providence, RI, (2010), 311–339. doi: 10.1090/crmp/050/24.

[7]

A. Bruce, Modular classes of $Q$-manifolds: a review and some applications, Archivum Mathematicum, 53 (2017), issue 4,203–219. doi: 10.5817/AM2017-4-203.

[8]

R. Campos, Homotopy Equivalence of Shifted Cotangent Bundles, Journal of Lie Theory, 29 (2019), 629-646. 

[9]

D. Calaque, R. Campos and J. Nuiten, Lie algebroids are curved Lie algebras, preprint, 2021, arXiv: 2103.10728.

[10]

R. Caseiro, The modular class of a Dirac map, J. Geom. Phys., 104 (2016), 19-29.  doi: 10.1016/j.geomphys.2016.01.007.

[11]

R. Caseiro and R.-L. Fernandes, The modular class of a Poisson map, Ann. Inst. Fourier, 63 (2013), no. 4, 1285–1329. doi: 10.5802/aif.2804.

[12]

A. Cattaneo and G. Felder, Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys., 69 (2004), 157-175.  doi: 10.1007/s11005-004-0609-7.

[13]

Z. ChenM. Stiénon and P. Xu, From Atiyah classes to homotopy Leibniz algebras, Comm. Math. Phys., 341 (2006), 309-349.  doi: 10.1007/s00220-015-2494-6.

[14]

S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser (2), 50 (1999), no. 200,417–436. doi: 10.1093/qjmath/50.200.417.

[15]

Y. Frégier and R. A. Juarez-Ojeda, Homotopy theory of singular foliations, preprint, 2018, arXiv: 1811.03078.

[16]

A. Gracia-Saz and K. Mackenzie, Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.  doi: 10.1007/s11005-009-0346-z.

[17]

A. Gracia-SazM. Jotz LeanK. Mackenzie and R. Mehta, Double Lie algebroids and representations up to homotopy, J. Homotopy Relat. Struct., 13 (2018), 287-319.  doi: 10.1007/s40062-017-0183-1.

[18]

J. Granåker, Unimodular $L_\infty$-algebras, preprint, 2008, arXiv: 0803.1763.

[19]

M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), no. 3,157–216. doi: 10.1023/B:MATH.0000027508.00421.bf.

[20]

Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids, in Quantization, Poisson brackets and beyond, 2001 (Manchester), Contemp. Math, 315, Amer. Math Soc., Providence, RI, 2002,213–233. doi: 10.1090/conm/315/05482.

[21]

Y. Kosmann-Schwarzbach and A. Weinstein, Relative modular classes of Lie algebroids, C. R. Math. Acad. Sci. Paris, 341 (2005), no. 8,509–514. doi: 10.1016/j.crma.2005.09.010.

[22]

Y. Kosmann-Schwarzbach, C. Laurent-Gengoux and A. Weinstein, Modular classes of Lie algebroid morphisms, Transformation Groups, 13 (2008), nos. 3-4,727–755. doi: 10.1007/s00031-008-9032-y.

[23]

J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in The mathematical heritage of Elie Cartan, Astérisque, numéro hors série, 1985,257-271.

[24]

M. Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math., 115 (1999), 71-113.  doi: 10.1023/A:1000664527238.

[25]

H. M. Khudaverdian and Th. Th. Voronov, Berezinians, exterior powers and recurrent sequences, Lett. Math. Phys., 74 (2005), no. 2,201–228. doi: 10.1007/s11005-005-0025-7.

[26]

H. M. Khudaverdian and Th. Th. Voronov, On odd Laplace operators, Lett. Math. Phys., 62 (2002), no. 2,127–142. doi: 10.1023/A:1021671812079.

[27]

H. M. Khudaverdian, Laplacians in odd symplectic geometry, Contemp. Math., 315 (2002), 199-212.  doi: 10.1090/conm/315/05481.

[28]

A. Kotov and T. Strobl, Generalizing geometry - algebroids and sigma models, in Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, 2010,209–262. doi: 10.4171/079-1/7.

[29]

T. Lada and J. Stacheff, Introduction to SH Lie algebras for physicists, Internat. J. Theoret. Phys., 32 (1993), no. 7, 1087–1103. doi: 10.1007/BF00671791.

[30]

T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. Algebra, 23 (1995), no. 6, 2147–2161. doi: 10.1080/00927879508825335.

[31]

C. Laurent-Gengoux C.M. Stiénon and P. Xu, Poincaré-Birkhoff-Witt isomorphisms and Kapranov dg-manifolds, Advances in Mathematics, 387 (2021), 107792.  doi: 10.1016/j.aim.2021.107792.

[32]

C. Laurent-GengouxS. Lavau and T. Strobl, The universal Lie $\infty$-algebroid of a singular foliation, Doc. Math., 25 (2020), 1571-1652. 

[33]

C. Laurent-Gengoux and R. Louis, Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty $-algebroid, J. of Algebra, 594 (2022), 1-53.  doi: 10.1016/j.jalgebra.2021.11.023.

[34]

S. Lavau, Lie $\infty$-algebroids and singular foliations, Ph.D thesis, Université Claude Bernard Lyon 1, 2017.

[35]

S. Lavau, The modular class of a singular foliation, preprint, 2022, arXiv: 2203.10861.

[36]

V. Salnikov, Graded geometry in gauge theories and beyond, J. Geom. Phys., 87 (2015), 422-431.  doi: 10.1016/j.geomphys.2014.07.001.

[37]

J.-L. Loday and B. Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer, 2012. doi: 10.1007/978-3-642-30362-3.

[38]

K. Mackenzie, A note on Lie algebroids which arise from groupoid actions, Cahiers Topologie Géom. Différentielle Catég., 28 (1987), no. 4,283–302.

[39]

K. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, (2005). doi: 10.1017/CBO9781107325883.

[40]

K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, (1987). doi: 10.1017/CBO9780511661839.

[41]

K. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, preprint, 1998, arXiv: math.DG/9808081. doi: 10.1090/S1079-6762-98-00050-X.

[42]

K. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.

[43]

R. Mehta, Lie algebroid modules and representations up to homotopy, Indagationes Mathematicae, 25 (2014), 1122-1134.  doi: 10.1016/j.indag.2014.07.013.

[44]

L. Ryvkin, $L_\infty$-algebras, in Observables and Symmetries of $n$-Plectic Manifolds, Springer, 2016, 3–28.

[45]

H. SatiU. Schreiber and J. Stasheff, Twisted differential string and fivebrane structures, Communications in Mathematical Physics, 315 (2012), 169-213.  doi: 10.1007/s00220-012-1510-3.

[46]

A.-Y. Vaintrob, Lie algebroids and homological vector fields, Uspekhi Mat. Nauk, 52 (1997), no. 2(314), 161–162. doi: 10.1070/RM1997v052n02ABEH001802.

[47]

Th. Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra, 202 (2005), no. 1-3,133–153. doi: 10.1016/j.jpaa.2005.01.010.

[48]

Th. Th.Voronov, $Q$-manifolds and higher analogs of Lie algebroids, In XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., 307, 2010,191–202.

[49]

Th. Th. Voronov, Q-manifolds and Mackenzie theory: an overview, preprint, 2007, arXiv: 0709.4232. doi: 10.1007/s00220-012-1568-y.

[50]

Th. Th. Voronov, Q-Manifolds and Mackenzie Theory, Commun. Math. Phys., 315 (2012), no 2,279–310. doi: 10.1007/s00220-012-1568-y.

[51]

A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394.  doi: 10.1016/S0393-0440(97)80011-3.

show all references

References:
[1]

C. Abad and M. Crainic, Representations up to homotopy of Lie algebroids, J. Reine Angew. Math., 663 (2012), 91-126.  doi: 10.1515/CRELLE.2011.095.

[2]

I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math., 626 (2009), 1-37.  doi: 10.1515/CRELLE.2009.001.

[3]

I. Androulidakis and M. Zambon, Holonomy transformations for singular foliations, Advances in Mathematics, 256 (2014), 348-397.  doi: 10.1016/j.aim.2014.02.003.

[4]

M.-J. Azimi, C. Laurent-Gengoux and J.-M.Nunes da Costa, Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids, Communications in Algebra, 46 (2018), issue 2,494–515. doi: 10.1080/00927872.2017.1288234.

[5]

G. Bonavolontà and N. Poncin, On the category of Lie $n$-algebroids, J. Geom. Phys., 73 (2013), 70-90.  doi: 10.1016/j.geomphys.2013.05.004.

[6]

J. Block, Duality and equivalence of module categories in noncommutative geometry, Memorial volume for R. Bott., CRM Proc. Lecture Notes, 50, Amer. Math. Soc., Providence, RI, (2010), 311–339. doi: 10.1090/crmp/050/24.

[7]

A. Bruce, Modular classes of $Q$-manifolds: a review and some applications, Archivum Mathematicum, 53 (2017), issue 4,203–219. doi: 10.5817/AM2017-4-203.

[8]

R. Campos, Homotopy Equivalence of Shifted Cotangent Bundles, Journal of Lie Theory, 29 (2019), 629-646. 

[9]

D. Calaque, R. Campos and J. Nuiten, Lie algebroids are curved Lie algebras, preprint, 2021, arXiv: 2103.10728.

[10]

R. Caseiro, The modular class of a Dirac map, J. Geom. Phys., 104 (2016), 19-29.  doi: 10.1016/j.geomphys.2016.01.007.

[11]

R. Caseiro and R.-L. Fernandes, The modular class of a Poisson map, Ann. Inst. Fourier, 63 (2013), no. 4, 1285–1329. doi: 10.5802/aif.2804.

[12]

A. Cattaneo and G. Felder, Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys., 69 (2004), 157-175.  doi: 10.1007/s11005-004-0609-7.

[13]

Z. ChenM. Stiénon and P. Xu, From Atiyah classes to homotopy Leibniz algebras, Comm. Math. Phys., 341 (2006), 309-349.  doi: 10.1007/s00220-015-2494-6.

[14]

S. Evens, J.-H. Lu and A. Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser (2), 50 (1999), no. 200,417–436. doi: 10.1093/qjmath/50.200.417.

[15]

Y. Frégier and R. A. Juarez-Ojeda, Homotopy theory of singular foliations, preprint, 2018, arXiv: 1811.03078.

[16]

A. Gracia-Saz and K. Mackenzie, Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.  doi: 10.1007/s11005-009-0346-z.

[17]

A. Gracia-SazM. Jotz LeanK. Mackenzie and R. Mehta, Double Lie algebroids and representations up to homotopy, J. Homotopy Relat. Struct., 13 (2018), 287-319.  doi: 10.1007/s40062-017-0183-1.

[18]

J. Granåker, Unimodular $L_\infty$-algebras, preprint, 2008, arXiv: 0803.1763.

[19]

M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66 (2003), no. 3,157–216. doi: 10.1023/B:MATH.0000027508.00421.bf.

[20]

Y. Kosmann-Schwarzbach and K. Mackenzie, Differential operators and actions of Lie algebroids, in Quantization, Poisson brackets and beyond, 2001 (Manchester), Contemp. Math, 315, Amer. Math Soc., Providence, RI, 2002,213–233. doi: 10.1090/conm/315/05482.

[21]

Y. Kosmann-Schwarzbach and A. Weinstein, Relative modular classes of Lie algebroids, C. R. Math. Acad. Sci. Paris, 341 (2005), no. 8,509–514. doi: 10.1016/j.crma.2005.09.010.

[22]

Y. Kosmann-Schwarzbach, C. Laurent-Gengoux and A. Weinstein, Modular classes of Lie algebroid morphisms, Transformation Groups, 13 (2008), nos. 3-4,727–755. doi: 10.1007/s00031-008-9032-y.

[23]

J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in The mathematical heritage of Elie Cartan, Astérisque, numéro hors série, 1985,257-271.

[24]

M. Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math., 115 (1999), 71-113.  doi: 10.1023/A:1000664527238.

[25]

H. M. Khudaverdian and Th. Th. Voronov, Berezinians, exterior powers and recurrent sequences, Lett. Math. Phys., 74 (2005), no. 2,201–228. doi: 10.1007/s11005-005-0025-7.

[26]

H. M. Khudaverdian and Th. Th. Voronov, On odd Laplace operators, Lett. Math. Phys., 62 (2002), no. 2,127–142. doi: 10.1023/A:1021671812079.

[27]

H. M. Khudaverdian, Laplacians in odd symplectic geometry, Contemp. Math., 315 (2002), 199-212.  doi: 10.1090/conm/315/05481.

[28]

A. Kotov and T. Strobl, Generalizing geometry - algebroids and sigma models, in Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, 2010,209–262. doi: 10.4171/079-1/7.

[29]

T. Lada and J. Stacheff, Introduction to SH Lie algebras for physicists, Internat. J. Theoret. Phys., 32 (1993), no. 7, 1087–1103. doi: 10.1007/BF00671791.

[30]

T. Lada and M. Markl, Strongly homotopy Lie algebras, Comm. Algebra, 23 (1995), no. 6, 2147–2161. doi: 10.1080/00927879508825335.

[31]

C. Laurent-Gengoux C.M. Stiénon and P. Xu, Poincaré-Birkhoff-Witt isomorphisms and Kapranov dg-manifolds, Advances in Mathematics, 387 (2021), 107792.  doi: 10.1016/j.aim.2021.107792.

[32]

C. Laurent-GengouxS. Lavau and T. Strobl, The universal Lie $\infty$-algebroid of a singular foliation, Doc. Math., 25 (2020), 1571-1652. 

[33]

C. Laurent-Gengoux and R. Louis, Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty $-algebroid, J. of Algebra, 594 (2022), 1-53.  doi: 10.1016/j.jalgebra.2021.11.023.

[34]

S. Lavau, Lie $\infty$-algebroids and singular foliations, Ph.D thesis, Université Claude Bernard Lyon 1, 2017.

[35]

S. Lavau, The modular class of a singular foliation, preprint, 2022, arXiv: 2203.10861.

[36]

V. Salnikov, Graded geometry in gauge theories and beyond, J. Geom. Phys., 87 (2015), 422-431.  doi: 10.1016/j.geomphys.2014.07.001.

[37]

J.-L. Loday and B. Vallette, Algebraic operads, Grundlehren der Mathematischen Wissenschaften, 346, Springer, 2012. doi: 10.1007/978-3-642-30362-3.

[38]

K. Mackenzie, A note on Lie algebroids which arise from groupoid actions, Cahiers Topologie Géom. Différentielle Catég., 28 (1987), no. 4,283–302.

[39]

K. Mackenzie, General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, (2005). doi: 10.1017/CBO9781107325883.

[40]

K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, (1987). doi: 10.1017/CBO9780511661839.

[41]

K. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, preprint, 1998, arXiv: math.DG/9808081. doi: 10.1090/S1079-6762-98-00050-X.

[42]

K. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.  doi: 10.1215/S0012-7094-94-07318-3.

[43]

R. Mehta, Lie algebroid modules and representations up to homotopy, Indagationes Mathematicae, 25 (2014), 1122-1134.  doi: 10.1016/j.indag.2014.07.013.

[44]

L. Ryvkin, $L_\infty$-algebras, in Observables and Symmetries of $n$-Plectic Manifolds, Springer, 2016, 3–28.

[45]

H. SatiU. Schreiber and J. Stasheff, Twisted differential string and fivebrane structures, Communications in Mathematical Physics, 315 (2012), 169-213.  doi: 10.1007/s00220-012-1510-3.

[46]

A.-Y. Vaintrob, Lie algebroids and homological vector fields, Uspekhi Mat. Nauk, 52 (1997), no. 2(314), 161–162. doi: 10.1070/RM1997v052n02ABEH001802.

[47]

Th. Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra, 202 (2005), no. 1-3,133–153. doi: 10.1016/j.jpaa.2005.01.010.

[48]

Th. Th.Voronov, $Q$-manifolds and higher analogs of Lie algebroids, In XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., 307, 2010,191–202.

[49]

Th. Th. Voronov, Q-manifolds and Mackenzie theory: an overview, preprint, 2007, arXiv: 0709.4232. doi: 10.1007/s00220-012-1568-y.

[50]

Th. Th. Voronov, Q-Manifolds and Mackenzie Theory, Commun. Math. Phys., 315 (2012), no 2,279–310. doi: 10.1007/s00220-012-1568-y.

[51]

A. Weinstein, The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23 (1997), 379-394.  doi: 10.1016/S0393-0440(97)80011-3.

[1]

Robert I. McLachlan, Ander Murua. The Lie algebra of classical mechanics. Journal of Computational Dynamics, 2019, 6 (2) : 345-360. doi: 10.3934/jcd.2019017

[2]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066

[3]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[4]

Juan Carlos Marrero, David Martín de Diego, Eduardo Martínez. Local convexity for second order differential equations on a Lie algebroid. Journal of Geometric Mechanics, 2021, 13 (3) : 477-499. doi: 10.3934/jgm.2021021

[5]

Robert L. Griess Jr., Ching Hung Lam. Groups of Lie type, vertex algebras, and modular moonshine. Electronic Research Announcements, 2014, 21: 167-176. doi: 10.3934/era.2014.21.167

[6]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239

[7]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453

[8]

Pengliang Xu, Xiaomin Tang. Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, 2021, 29 (4) : 2771-2789. doi: 10.3934/era.2021013

[9]

Mark Wilkinson. A Lie algebra-theoretic approach to characterisation of collision invariants of the Boltzmann equation for general convex particles. Kinetic and Related Models, 2022, 15 (2) : 283-315. doi: 10.3934/krm.2022008

[10]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, 2021, 29 (3) : 2445-2456. doi: 10.3934/era.2020123

[11]

Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39

[12]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[13]

Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007

[14]

André Caldas, Mauro Patrão. Entropy of endomorphisms of Lie groups. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1351-1363. doi: 10.3934/dcds.2013.33.1351

[15]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517

[16]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124

[17]

Marco Zambon. Holonomy transformations for Lie subalgebroids. Journal of Geometric Mechanics, 2021, 13 (3) : 517-532. doi: 10.3934/jgm.2021016

[18]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, 2021, 29 (5) : 2945-2957. doi: 10.3934/era.2021020

[19]

K. C. H. Mackenzie. Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids. Electronic Research Announcements, 1998, 4: 74-87.

[20]

Johannes Huebschmann. On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras. Journal of Geometric Mechanics, 2021, 13 (3) : 385-402. doi: 10.3934/jgm.2021009

2021 Impact Factor: 0.737

Article outline

[Back to Top]