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数学进展 - 2020, Vol. 49(6): 641-674
综述文章
李2-代数综述
A Review of Lie 2-algebras

郎红蕾1, 刘张炬2
LANG Honglei1, LIU Zhangju2

1. 中国农业大学应用数学系, 北京, 100083;
2. 北京大学数学科学学院, 北京, 100871
1. Department of Applied Mathematics, China Agricultural University, Beijing, 100083, P. R. China;
2. School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.15in

收稿日期: 2020-05-07
出版日期: 2020-11-17
2020, Vol. 49(6): 641-674
DOI: 10.11845/sxjz.2020003a


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摘要 首先分别从范畴化和L-无穷代数两个不同角度回顾了李2-代数的两种等价定义. 接着给出了从4种几何结构中产生的李2-代数. 进一步, 阐述了李2-代数的上同调理论并具体分析了低阶情况. 最后讨论了严格李2-代数到严格李2-群的积分.
关键词 李2-代数范畴化上同调李2-群    
Abstract:We first recall two equivalent definitions of Lie 2-algebras, categorification of Lie algebras and 2-term $L_{\infty}$-algebras. Then we present four different kinds of Lie 2-algebras from 2-plectic manifolds, Courant algebroids, homotopy Poisson manifolds and affine multivector fields on a Lie groupoid respectively. Moreover, we recall the cohomology theory of Lie 2-algebras and analyze its lower degree cases. The integration of strict Lie 2-algebras to strict Lie 2-groups is also discussed.
Key wordsLie 2-algebras    categorification    cohomology    Lie 2-groups
PACS:  O152.5  
[1] Angulo C.,A cohomology theory for Lie 2-algebras and Lie 2-groups, Ph.D. Thesis, S~ao Paulo: Instituto de Matem'atica e Estat'istica, Universidade de S~ao Paulo, 2018.<br /> [2] Baez, J.C. and Crans, A.S., Higher-dimensional algebra, VI, Lie 2-algebras, Theory Appl. Categ., 2014, 12: 492-538.<br /> [3] Baez J.C., Hoffnung A.E. and Rogers C.L., Categorified symplectic geometry and the classical string, Comm. Math. Phys., 2010, 293(3): 701-725.<br /> [4] Baez, J.C. and Lauda, A.D., Higher-dimensional algebra, V, 2-groups, Theory Appl. Categ., 2014, 12: 423-491.<br /> [5] Baez, J.C. and Schreiber, U., Higher gauge theory, In: Categories in Algebra, Geometry and Mathematical Physics, Contemp. Math., Vol. 431, Providence, RI: AMS, 2007, 7-30.<br /> [6] Baez J.C., Stevenson D., Crans A.S. and Schreiber U., From loop groups to 2-groups, Homology Homotopy Appl., 2007, 9(2): 101-135.<br /> [7] Bai C.M., Sheng Y.H. and Zhu C.C., Lie 2-bialgebras, Comm. Math. Phys., 2013, 320(1): 149-172.<br /> [8] Berwick-Evans, D. and Lerman, E., Lie 2-algebras of vector fields, 2016, arXiv:1609.03944.<br /> [9] Bonechi F., Ciccoli N., Laurent-Gengoux C. and Xu P., Shifted Poisson structures on differentiable stacks,2018, arXiv:1803.06685.<br /> [10] Cai X.W., Liu Z.J. and Xiang M.S., Cohomology of hemistrict Lie 2-algebras, Comm. Algebra, 2020, 48(8): 3315-3341.<br /> [11] Callies M.,Fr'egier, Y., Rogers, C.L. and Zambon, M., Homotopy moment maps, Adv. Math., 2016, 303: 954-1043.<br /> [12] Cattaneo, A.S. and Felder, G., Relative formality theorem and quantisation of coisotropic submanifolds, Adv. Math., 2007, 208(2): 521-548.<br /> [13] Chen S.H., Sheng Y.H. and Zheng Z.J., Non-abelian extensions of Lie 2-algebras, Sci. China Math., 2012, 55(8): 1655-1668.<br /> [14] Chen Z.,Sti'enon, M. and Xu. P., Weak Lie 2-bialgebras, [J]. Geom. Phys., 2013, 68: 59-68.<br /> [15] Chen Z.,Sti'enon, M. and Xu. P., Poisson 2-groups, [J]. Differential Geom., 2013, 94(2): 209-240.<br /> [16] Courant T.J.,Dirac manifolds, Trans. Amer. Math. Soc., 1990, 319(2): 631-661.<br /> [17] Crainic, M. and Fernandes, R.L., Integrability of Lie brackets, Ann. of Math. (2), 2003, 157(2): 575-620.<br /> [18] Dorfman I.,Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Science: Theory and Applications, Chichester: John Wiley & Sons, 1993.<br /> [19] Doubek, M. and Lada, T., Homotopy derivations, [J]. Homotopy Relat. Struct., 2016, 11(3): 599-630.<br /> [20] Fr'egier, Y., Laurent-Gengoux, C. and Zambon, M., A cohomological framework for homotopy moment maps, [J]. Geom. Phys., 2015, 97: 119-132.<br /> [21] Gerstenhaber M.,A uniform cohomology theory for algebras, Proc. Nat. Acad. Sci. U.S.A., 1964, 51: 626-629.<br /> [22] Getzler E.,Lie theory for nilpotent $L_{\infty}$-algebras, Ann. of Math. (2), 2009, 170(1): 271-301.<br /> [23] Gualtieri M.,Generalized complex geometry, Ann. of Math. (2), 2011, 174(1): 75-123.<br /> [24] Henriques A.,Integrating $L_{\infty}$-algebras, Compos. Math., 2008, 144(4): 1017-1045.<br /> [25] Iglesias-Ponte, D., Laurent-Gengoux, C. and Xu, P., Universal lifting theorem and quasi-Poisson groupoids, J. Eur. Math. Soc. (JEMS), 2012, 14(3): 681-731.<br /> [26] Kelly, G.M. and Street, R., Review of the elements of 2-categories, In: Category Seminar (Proc. Sem., Sydney, 1972/1973), Lecture Notes in Math., Vol. 420, Berlin: Springer, 1974, 75-103.<br /> [27] Khudaverdian, H. and Voronov, T.T., Higher Poisson brackets and differential forms, In: Geometric Methods in Physics, AIP Conf. Proc., Vol. 1079, Melville, NY: Amer. Inst. Phys., 2008, 203-215.<br /> [28] Kinyon, M.K. and Weinstein, A., Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. [J]. Math., 2001, 123(3): 525-550.<br /> [29] Kravchenko O.,Strongly homotopy Lie bialgebras and Lie quasi-bialgebras, Lett. Math. Phys., 2007, 81(1): 19-40.<br /> [30] Lada, T. and Markl, M., Strongly homotopy Lie algebras, Comm. Algebra, 1995, 23(6): 2147-2161.<br /> [31] Lada, T. and Stasheff, J., Introduction to SH Lie algebras for physicists, Internat. [J]. Theoret. Phys., 1993, 32(7): 1087-1103.<br /> [32] Lang, H.L. and Liu, Z.J., Crossed modules for Lie 2-algebras, Appl. Categ. Structures, 2016, 24(1): 53-78.<br /> [33] Lang H.L., Liu Z.J. and Sheng Y.H., Integration of derivations for Lie 2-algebras, Transform. Groups, 2016, 21(1): 129-152.<br /> [34] Lang H.L., Liu Z.J. and Sheng Y.H., Affine structures on Lie groupoids, Pacific [J]. Math., 2020, 307(2): 353-382.<br /> [35] Lang H.L., Sheng Y.H. and Xu X.M., Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids, Lett. Math. Phys., 2017, 107(5): 861-885.<br /> [36] Liu Z.J., Sheng Y.H. and Xu X.M., The Pontryagin class for pre-Courant algebroids, [J]. Geom. Phys., 2016, 104: 148-162.<br /> [37] Liu Z.J., Sheng Y.H. and Zhang T., Deformations of Lie 2-algebras, [J]. Geom. Phys., 2014, 86: 66-80.<br /> [38] Liu Z.J., Weinstein A. and Xu P., Manin triples for Lie bialgebroids, [J]. Differential Geom., 1997, 45(3): 547-574.<br /> [39] Lu J.H.,Multiplicative and affine Poisson structures on Lie groups, Ph.D. Thesis, Berkeley, CA: University of California, Berkeley, 1990.<br /> [40] Mackenzie K.C.H., General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge: Cambridge University Press, 2005.<br /> [41] Mammadova L. and Zambon M., Lie 2-algebra moment maps in multisymplectic geometry, Differential Geom. Appl., 2020, 70: 101631, 22 pp.<br /> [42] Markl M.,A cohomology theory for $A(m)$-algebras and applications, J. Pure Appl. Algebra, 1992, 83(2): 141-175.<br /> [43] Mill`es, J., Andr'e-Quillen cohomology of algebras over an operad, Adv. Math., 2011, 226(6): 5120-5164.<br /> [44] Noohi B.,Integrating morphisms of Lie 2-algebras, Compos. Math., 2013, 149(2): 264-294.<br /> [45] Ortiz C. and Waldron J., On the Lie 2-algebra of sections of an $\mathcal{LA}$-groupoid, J. Geom. Phys., 2019, 145: 103474, 34 pp.<br /> [46] Penkava, M., $L_{\infty}$-algebras and their cohomology, 1995, arXiv:q-alg/9512014.<br /> [47] Rogers C.L.,$L_{\infty}$-algebras from multisymplectic geometry, Lett. Math. Phys. 2012, 100(1): 29-50.<br /> [48] Rogers C.L.,2-plectic geometry, Courant algebroids, and categorified prequantization, [J]. Symplectic Geom., 2013, 11(1): 53-91.<br /> [49] Roytenberg D.,Courant algebroids, derived brackets and even symplectic supermanifolds, Ph.D. Thesis, Berkeley, CA: University of California, Berkeley. 1999.<br /> [50] Roytenberg D.,On the structure of graded symplectic manifolds and Courant algebroids, In: Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Providence, RI: AMS, 2002, 169-185.<br /> [51] Roytenberg D.,On weak Lie 2-algebras, In: XXVI Workshop on Geometrical Methods in Physics, AIP Conf. Proc., Vol. 956, Melville, NY: Amer. Inst. Phys., 2007, 180-198.<br /> [52] Roytenberg, D. and Weinstein, A., Courant algebroids and strongly homotopy Lie algebras, Lett. Math. Phys., 1998, 46(1): 81-93.<br /> [53] Schlessinger, M. and Stasheff, J., The Lie algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra, 1985, 38(2/3): 313-322.<br /> [54] \v{S}evera, P. and Weinstein, A., Poisson geometry with a 3-form background, Progr. Theoret. Phys. Suppl., 2001, 144: 145-154.<br /> [55] Sheng, Y.H. and Liu, Z.J., From Leibniz algebras to Lie 2-algebras, Algebr. Represent. Theory, 2016, 19(1): 1-5.<br /> [56] Sheng, Y.H. and Zhu, C.C., Semidirect products of representations up to homotopy, Pacific [J]. Math., 2011, 249(1): 211-236.<br /> [57] Sheng Y.H. and Zhu C.C., Integration of semidirect product Lie 2-algebras, Int. J. Geom. Methods Mod. Phys., 2012, 9(5): 1250043, 31 pp.<br /> [58] Sheng, Y.H. and Zhu, C.C., Integration of Lie 2-algebras and their morphisms, Lett. Math. Phys., 2012, 102(2): 223-244.<br /> [59] Voronov T.,Graded manifolds and Drinfeld doubles for Lie bialgebroids, In: Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Providence, RI: AMS, 2002, 131-168.<br /> [60] Wagemann F.,On Lie algebra crossed modules, Comm. Algebra, 2006, 34(5): 1699-1722.<br /> [61] Waldorf K.,A global perspective to connections on principal 2-bundles, Forum. Math., 2018, 30(4): 809-843.<br /> [62] Weinstein A.,Affine Poisson structures, Internat. [J]. Math., 1990, 1(3): 343-360.<br /> [63] Weinstein A.,Omni-Lie algebras, S$\bar{u}$rikaisekikenky$\bar{u}$sho K$\bar{o}$ky$\bar{u}$roku, 2000, 1176: 95-102.
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