September  2021, 13(3): 385-402. doi: 10.3934/jgm.2021009

On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras

Université de Lille - Sciences et Technologies, Département de Mathématiques, CNRS-UMR 8524, Labex CEMPI (ANR-11-LABX-0007-01), 59655 Villeneuve d'Ascq Cedex, France

Dedicated to the memory of Kirill Mackenzie

Received  March 2021 Published  September 2021 Early access  May 2021

This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmüller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra.

Citation: 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
References:
[1]

R. Baer, Erweiterung von Gruppen und ihren Isomorphismen, Math. Z., 38 (1934), 375-416.  doi: 10.1007/BF01170643.

[2]

R. Baer, Groups with abelian central quotient group, Trans. Amer. Math. Soc., 44 (1938), 357-386.  doi: 10.1090/S0002-9947-1938-1501972-1.

[3]

J. C. BaezD. StevensonA. S. Crans and U. Schreiber, From loop groups to 2-groups, Homology Homotopy Appl., 9 (2007), 101-135.  doi: 10.4310/HHA.2007.v9.n2.a4.

[4]

R. Brauer, Über die algebraische Struktur von Schiefkörpern, J. Reine Angew. Math., 166 (1932), 241-252.  doi: 10.1515/crll.1932.166.241.

[5]

R. Brown, On the second relative homotopy group of an adjunction space: An exposition of a theorem of J. H. C. Whitehead, J. London Math. Soc. (2), 22 (1980), 146-152. doi: 10.1112/jlms/s2-22.1.146.

[6]

R. Brown, P. J. Higgins and R. Sivera, Nonabelian Algebraic Topology, vol. 15 of EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2011. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids. With contributions by Christopher D. Wensley and Sergei V. Soloviev. doi: 10.4171/083.

[7] R. Brown and J. Huebschmann, Identities among relations, in Low-Dimensional Topology (Bangor, 1979), vol. 48 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 1982. 
[8]

R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Topologie Géom. Différentielle, 17 (1976), 343-362. 

[9]

U. Bruzzo, I. Mencattini, V. N. Rubtsov and P. Tortella, Nonabelian holomorphic Lie algebroid extensions, Internat. J. Math., 26 (2015), 1550040, 26 pp. doi: 10.1142/S0129167X15500408.

[10]

J. -L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, vol. 107 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-0-8176-4731-5.

[11]

L. Calabi, Sur les extensions des groupes topologiques, Ann. Mat. Pura Appl., 32 (1951), 295-370.  doi: 10.1007/BF02417964.

[12]

E. Cartan, Sur les nombres de Betti des espaces de groupes clos, C. R. Acad. Sci., Paris, 187 (1928), 196-198. 

[13]

H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, in Colloque de Topologie (Espaces Fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, 57-71.

[14]

H. Cartan, Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, in Colloque de Topologie (Espaces Fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, 15-27.

[15]

H. Cartan and S. Eilenberg, Homological Algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999., With an appendix by David A. Buchsbaum, Reprint of the 1956 original.

[16]

P. Cartier, Effacement dans la cohomologie des algèbres de Lie. Semin. Bourbaki, Vol. 3, Exp. No. 116, Soc. Math. France, Paris, 1995, 161-167.

[17]

S. C. Chang, On Jacobi identity, Acta Math. Sinica, 4 (1954), 365-379. 

[18]

J. Duskin, Simplicial Methods and the Interpretation of "Triple" Cohomology, Mem. Amer. Math. Soc., 3 (1975), v+135 pp. doi: 10.1090/memo/0163.

[19]

S. Eilenberg and S. MacLane, Group extensions and homology, Ann. of Math. (2), 43 (1942), 757-831. doi: 10.2307/1968966.

[20]

S. Eilenberg and S. MacLane, Cohomology theory in abstract groups. I, Ann. of Math. (2), 48 (1947), 51-78. doi: 10.2307/1969215.

[21]

S. Eilenberg and S. MacLane, Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel, Ann. of Math. (2), 48 (1947), 326-341. doi: 10.2307/1969174.

[22]

S. Eilenberg and S. MacLane, Cohomology and Galois theory. I. Normality of algebras and Teichmüller's cocycle, Trans. Amer. Math. Soc., 64 (1948), 1-20.  doi: 10.2307/1990556.

[23]

T. M. Fiore, Pseudo algebras and pseudo double categories, J. Homotopy Relat. Struct., 2 (2007), 119-170. 

[24]

A. Fröhlich and C. T. C. Wall, Equivariant Brauer groups, in Quadratic forms and their Applications (Dublin, 1999), vol. 272 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2000, 57-71. doi: 10.1090/conm/272/04397.

[25]

M. Gerstenhaber, A uniform cohomology theory for algebras, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 626-629.  doi: 10.1073/pnas.51.4.626.

[26]

M. Gerstenhaber, On the deformation of rings and algebras. II, Ann. of Math. (2), 84 (1966), 1-19. doi: 10.2307/1970528.

[27]

S. I. Goldberg, Extensions of Lie algebras and the third cohomology group, Canad. J. Math., 5 (1953), 470-476.  doi: 10.4153/cjm-1953-054-8.

[28]

B. Grossman, The meaning of the third cocycle in the group cohomology of nonabelian gauge theories, Phys. Lett. B, 160 (1985), 94-100.  doi: 10.1016/0370-2693(85)91472-8.

[29]

M. Hall, A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc., 1 (1950), 575-581.  doi: 10.1090/S0002-9939-1950-0038336-7.

[30]

P. Hall, A contribution to the theory of groups of prime-power order, Proc. Lond. Math. Soc., 36 (1934), 29-95.  doi: 10.1112/plms/s2-36.1.29.

[31]

K. HauserF. HerrlichM. KneserH. OpolkaN. Schappacher and E. Scholz, Oswald Teichmüller-leben und werk, Jahresber. Deutsch. Math.-Verein., 94 (1992), 1-39. 

[32]

J.-C. Herz, Pseudo-algèbres de Lie. I, C. R. Acad. Sci. Paris, 236 (1953), 1935-1937. 

[33]

J.-C. Herz, Pseudo-algèbres de Lie. II, C. R. Acad. Sci. Paris, 236 (1953), 2289-2291. 

[34]

P. J. Higgins and K. C. H. Mackenzie, Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures, Math. Proc. Cambridge Philos. Soc., 114 (1993), 471-488.  doi: 10.1017/S0305004100071760.

[35]

P. Hilton, Reminiscences and reflections of a codebreaker, in, Coding Theory and Cryptography (Annapolis, MD, 1998), Springer, Berlin, 2000, 1-8.

[36]

P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc., 30 (1955), 154-172.  doi: 10.1112/jlms/s1-30.2.154.

[37]

P. J. Hilton, Note on quasi-Lie rings, Fund. Math., 43 (1956), 230-237.  doi: 10.4064/fm-43-2-230-237.

[38]

G. Hochschild, Lie algebra kernels and cohomology, Amer. J. Math., 76 (1954), 698-716.  doi: 10.2307/2372712.

[39]

G. Hochschild, Simple algebras with purely inseparable splitting fields of exponent $1$, Trans. Amer. Math. Soc., 79 (1955), 477-489.  doi: 10.2307/1993043.

[40]

G. Hochschild and J.-P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc., 74 (1953), 110-134.  doi: 10.1090/S0002-9947-1953-0052438-8.

[41]

D. F. Holt, An interpretation of the cohomology groups $H^n(G, M)$, J. Algebra, 60 (1979), 307-318.  doi: 10.1016/0021-8693(79)90084-X.

[42]

J. Huebschmann, Verschränkte n-fache Erweiterungen von Gruppen und Cohomologie, 1977. Diss. Math. ETH Zürich, Nr. 5999. Ref. : Eckmann, B.; Korref. : Stammbach, U.

[43] J. Huebschmann, Cohomology theory of aspherical groups and of small cancellation groups, in Homological Group Theory (Proc. Sympos., Durham, 1977), vol. 36 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 1979. 
[44]

J. Huebschmann, Cohomology theory of aspherical groups and of small cancellation groups, J. Pure Appl. Algebra, 14 (1979), 137-143.  doi: 10.1016/0022-4049(79)90003-3.

[45]

J. Huebschmann, Crossed $n$-fold extensions of groups and cohomology, Comment. Math. Helv., 55 (1980), 302-313.  doi: 10.1007/BF02566688.

[46]

J. Huebschmann, Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence, J. Algebra, 72 (1981), 296-334.  doi: 10.1016/0021-8693(81)90296-9.

[47]

J. Huebschmann, Group extensions, crossed pairs and an eight term exact sequence, J. Reine Angew. Math., 321 (1981), 150-172.  doi: 10.1515/crll.1981.321.150.

[48]

J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math., 408 (1990), 57-113. https://arxiv.org/abs/1303.3903. doi: 10.1515/crll. 1990.408.57.

[49]

J. Huebschmann, On the quantization of Poisson algebras, in Symplectic Geometry and Mathematical Physics (Aix-en-Provence, 1990), vol. 99 of Progr. Math., Birkhäuser Boston, Boston, MA, 1991, 204-233.

[50]

J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Ann. Inst. Fourier (Grenoble), 48(1998), 425-440. https://arxiv.org/abs/dg-ga/9704005.

[51]

J. Huebschmann, Twilled Lie-Rinehart algebras and differential Batalin-Vilkovisky algebras, (1998). https://arxiv.org/abs/math/9811069.

[52]

J. Huebschmann, Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras, in Poisson Geometry (Warsaw, 1998), vol. 51 of Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 2000, 87-102. https://arxiv.org/abs/1303.3414.

[53]

J. Huebschmann, Lie-Rinehart algebras, descent, and quantization, in Galois Theory, Hopf Algebras, and Semiabelian Categories, vol. 43 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 2004, 295-316. https://arxiv.org/abs/math/0303016. doi: 10.1090/memo/0814.

[54]

J. Huebschmann, Singular Poisson-Kähler geometry of stratified Kähler spaces and quantization, in Geometry and Quantization, vol. 19 of Trav. Math., Univ. Luxemb., Luxembourg, 2011, 27-63. https://arxiv.org/abs/1103.1584.

[55]

J. Huebschmann, Braids and crossed modules, J. Group Theory, 15 (2012), 57-83. https://arxiv.org/abs/0904.3895. doi: 10.1515/JGT. 2011.095.

[56]

J. Huebschmann, Normality of algebras over commutative rings and the Teichmüller class. I, J. Homotopy Relat. Struct., 13 (2018), 1-70. https://arxiv.org/abs/1512.07264. doi: 10.1007/s40062-017-0173-3.

[57]

J. Huebschmann, Normality of algebras over commutative rings and the Teichmüller class. II, J. Homotopy Relat. Struct., 13 (2018), 71-125. https://arxiv.org/abs/1512.07264. doi: 10.1007/s40062-017-0174-2.

[58]

N. Jacobson, An extension of Galois theory to non-normal and non-separable fields, Amer. J. Math., 66 (1944), 1-29.  doi: 10.2307/2371892.

[59]

V. F. R. Jones, An invariant for group actions, in Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), vol. 725 of Lecture Notes in Math., Springer, Berlin, 1979, 237-253.

[60]

C. Kassel and J. -L. Loday, Extensions centrales d'algèbres de Lie, Ann. Inst. Fourier (Grenoble), 32 (1982), 119-142 (1983).

[61]

J. -L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque, The Mathematical Heritage of Élie Cartan (Lyon, 1984). (1985), 257-271.

[62]

J. -L. Koszul, Lectures on Fibre Bundles and Differential Geometry, vol. 20 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-02503-1.

[63]

A. G. Kurosch, Gruppentheorie. I, Aus dem Russischen übersetzt von Reinhard Strecker und Heidemarie Strecker. Herausgegeben von Reinhard Strecker. Zweite, überarbeitete und erweiterte Auflage. Mathematische Lehrbücher und Monographien, I. Abteilung, Mathematische Lehrbücher, Band III/I, Akademie-Verlag, Berlin, 1970.

[64]

A. G. Kurosch, Gruppentheorie. II, Akademie-Verlag, Berlin, 1972., Aus dem Russischen übersetzt von Reinhard Strecker und Heidemarie Strecker. Herausgegeben von Reinhard Strecker, Zweite, überarbeitete und erweiterte Auflage, Mathematische Lehrbücher und Monographien, I. Abteilung, Mathematische Lehrbücher, Band III/II.

[65]

C. R. Leedham-Green and S. McKay, Baer-invariants, isologism, varietal laws and homology, Acta Math., 137 (1976), 99-150.  doi: 10.1007/BF02392415.

[66]

J.-L. Loday, Cohomologie et groupe de Steinberg relatifs, J. Algebra, 54 (1978), 178-202.  doi: 10.1016/0021-8693(78)90025-X.

[67]

R. C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2), 52 (1950), 650-665. doi: 10.2307/1969440.

[68] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, vol. 124 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.
[69]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.

[70] K. C. H. Mackenzie, General Theory of Lie groupoids and Lie Algebroids, vol. 213 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[71]

S. MacLane, Cohomology theory in abstract groups. III. Operator homomorphisms of kernels, Ann. of Math. (2), 50 (1949), 736-761. doi: 10.2307/1969561.

[72]

S. MacLane, Homology. Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin, first ed., 1967.

[73]

S. MacLane, Historical note, J. Algebra, 60 (1979), 319-320. Appendix to [41]. doi: 10.1016/0021-8693(79)90085-1.

[74]

S. MacLane and J. H. C. Whitehead, On the $3$-type of a complex, Proc. Nat. Acad. Sci. U. S. A., 36 (1950), 41-48.  doi: 10.1073/pnas.36.1.41.

[75]

W. Magnus, Über Beziehungen zwischen höheren Kommutatoren, J. Reine Angew. Math., 177 (1937), 105-115.  doi: 10.1515/crll.1937.177.105.

[76]

W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Interscience Publishers [John Wiley & Sons, Inc. ], New York-London-Sydney, 1966.

[77] W. Metzler, Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen, in Homological Group Theory (Proc. Sympos., Durham, 1977), vol. 36 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 1979. 
[78]

M. Mori, On the three-dimensional cohomology group of Lie algebras, J. Math. Soc. Japan, 5 (1953), 171-183.  doi: 10.2969/jmsj/00520171.

[79]

M. Nakaoka and H. Toda, On Jacobi identity for Whitehead products, J. Inst. Polytech. Osaka City Univ. Ser. A, 5 (1954), 1-13. 

[80]

K.-H. Neeb, Non-abelian extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), 57 (2007), 209-271. 

[81]

W. Nichols and B. Weisfeiler, Differential formal groups of J. F. Ritt, Amer. J. Math., 104 (1982), 943-1003.  doi: 10.2307/2374080.

[82]

E. Noether, Hyperkomplexe Größen und Darstellungstheorie, Math. Z., 30 (1929), 641-692.  doi: 10.1007/BF01187794.

[83]

E. Noether, Hyperkomplexe Größen und Darstellungstheorie in arithmetischer Auffassung, Atti Congresso Bologna (1928), 2 (1930), 71-73. 

[84]

K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math., 76 (1954), 33-65.  doi: 10.2307/2372398.

[85]

K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (1990), 129-146. 

[86]

R. S. Palais, The cohomology of Lie rings, in Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 130-137.

[87]

C. D. Papakyriakopoulos, Attaching $2$-dimensional cells to a complex, Ann. of Math. (2), 78 (1963), 205-222. doi: 10.2307/1970340.

[88]

R. Peiffer, Über Identitäten zwischen Relationen, Math. Ann., 121 (1949), 67-99.  doi: 10.1007/BF01329617.

[89]

J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245-A248.

[90]

J. G. Ratcliffe, The theory of crossed modules with applications to cohomology of groups and combinatorial homotopy theory, ProQuest LLC, Ann Arbor, MI, 1977., Thesis (Ph. D. )-University of Michigan.

[91]

K. Reidemeister, Knoten und Gruppen, Abh. Math. Semin. Univ. Hamb., 5 (1926), 7-23.  doi: 10.1007/BF02952506.

[92]

K. Reidemeister, Knotentheorie, vol. 1, Springer-Verlag, Berlin, 1932.

[93]

K. Reidemeister, Homotopiegruppen von Komplexen, Abh. Math. Semin. Univ. Hamb., 10 (1934), 211-215.  doi: 10.1007/BF02940675.

[94]

K. Reidemeister, Topologie der Polyeder und kombinatorische Topologie der Komplexe. (Math. u. ihre Anwendungen in Monogr. u. Lehrbüchern. 17). Leipzig: Akad. Verlagsges. mbH IX, 196 S., 69 Fig., 1938.

[95]

K. Reidemeister, Über Identitäten von Relationen, Abh. Math. Sem. Univ. Hamburg, 16 (1949), 114-118.  doi: 10.1007/BF03343521.

[96]

G. S. Rinehart, Differential forms for general commutative algebras, Trans. Amer. Math. Soc., 108 (1963), 195-222.  doi: 10.1090/S0002-9947-1963-0154906-3.

[97]

G. S. Rinehart, Satellites and cohomology, J. Algebra, 12 (1969), 295-329.  doi: 10.1016/0021-8693(69)90032-5.

[98]

J. F. Ritt, Systems of Differential Equations. I. Theory of Ideal, Amer. J. Math., 60 (1938), 535-548.  doi: 10.2307/2371594.

[99]

J. F. Ritt, Associative differential operations, Ann. of Math. (2), 51 (1950), 756-765. doi: 10.2307/1969379.

[100]

J. F. Ritt, Differential Algebra, American Mathematical Society Colloquium Publications, Vol. XXXIII, American Mathematical Society, New York, N. Y., 1950.

[101]

J. F. Ritt, Differential groups and formal Lie theory for an infinite number of parameters, Ann. of Math. (2), 52 (1950), 708-726. doi: 10.2307/1969444.

[102]

H. Samelson, A connection between the Whitehead and the Pontryagin product, Amer. J. Math., 75 (1953), 744-752.  doi: 10.2307/2372549.

[103]

O. Schreier, Über die Erweiterung von Gruppen I, Monatsh. Math. Phys., 34 (1926), 165-180.  doi: 10.1007/BF01694897.

[104]

A. J. Sieradski, Combinatorial isomorphisms and combinatorial homotopy equivalences, J. Pure Appl. Algebra, 7 (1976), 59-95.  doi: 10.1016/0022-4049(76)90067-0.

[105]

P. A. Smith, The complex of a group relative to a set of generators. I, Ann. of Math. (2), 54 (1951), 371-402. doi: 10.2307/1969538.

[106]

J. D. Stasheff, Continuous cohomology of groups and classifying spaces, Bull. Amer. Math. Soc., 84 (1978), 513-530.  doi: 10.1090/S0002-9904-1978-14488-7.

[107]

H. Suzuki, A product in homotopy theory, Tohoku Math. J., 6 (1954), 78-88.  doi: 10.2748/tmj/1178245238.

[108]

O. Teichmüller, Über die sogenannte nichtkommutative Galoissche Theorie und die Relation $\xi_\lambda, \mu, \nu\xi_\lambda, \mu\nu, \pi\xi^\lambda_\mu, \nu, \pi = \xi_\lambda, \mu, \nu \pi\xi_\lambda, \mu, \nu, \pi$, Deutsche Math., 5 (1940), 138-149. 

[109]

A. M. Turing, The extensions of a group, Compositio Math., 5 (1938), 357-367. 

[110] H. Uehara and W. S. Massey, The Jacobi identity for Whitehead products, in Algebraic Geometry and Topology. A symposium in Honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957. 
[111]

G. W. Whitehead, On mappings into group-like spaces, Comment. Math. Helv., 28 (1954), 320-328.  doi: 10.1007/BF02566938.

[112]

J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2), 42 (1941), 409-428. doi: 10.2307/1968907.

[113]

J. H. C. Whitehead, Note on a previous paper entitled "On adding relations to homotopy groups. ", Ann. of Math. (2), 47 (1946), 806-810. doi: 10.2307/1969237.

[114]

J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc., 55 (1949), 453-496.  doi: 10.1090/S0002-9904-1949-09213-3.

[115]

E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math., 177 (1937), 152-160. doi: 10.1515/crll. 1937.177.152.

[116]

Y. C. Wu, $H^{3}(G, A)$ and obstructions of group extensions, J. Pure Appl. Algebra, 12 (1978), 93-100.  doi: 10.1016/0022-4049(78)90023-3.

show all references

References:
[1]

R. Baer, Erweiterung von Gruppen und ihren Isomorphismen, Math. Z., 38 (1934), 375-416.  doi: 10.1007/BF01170643.

[2]

R. Baer, Groups with abelian central quotient group, Trans. Amer. Math. Soc., 44 (1938), 357-386.  doi: 10.1090/S0002-9947-1938-1501972-1.

[3]

J. C. BaezD. StevensonA. S. Crans and U. Schreiber, From loop groups to 2-groups, Homology Homotopy Appl., 9 (2007), 101-135.  doi: 10.4310/HHA.2007.v9.n2.a4.

[4]

R. Brauer, Über die algebraische Struktur von Schiefkörpern, J. Reine Angew. Math., 166 (1932), 241-252.  doi: 10.1515/crll.1932.166.241.

[5]

R. Brown, On the second relative homotopy group of an adjunction space: An exposition of a theorem of J. H. C. Whitehead, J. London Math. Soc. (2), 22 (1980), 146-152. doi: 10.1112/jlms/s2-22.1.146.

[6]

R. Brown, P. J. Higgins and R. Sivera, Nonabelian Algebraic Topology, vol. 15 of EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2011. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids. With contributions by Christopher D. Wensley and Sergei V. Soloviev. doi: 10.4171/083.

[7] R. Brown and J. Huebschmann, Identities among relations, in Low-Dimensional Topology (Bangor, 1979), vol. 48 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 1982. 
[8]

R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers Topologie Géom. Différentielle, 17 (1976), 343-362. 

[9]

U. Bruzzo, I. Mencattini, V. N. Rubtsov and P. Tortella, Nonabelian holomorphic Lie algebroid extensions, Internat. J. Math., 26 (2015), 1550040, 26 pp. doi: 10.1142/S0129167X15500408.

[10]

J. -L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, vol. 107 of Progress in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-0-8176-4731-5.

[11]

L. Calabi, Sur les extensions des groupes topologiques, Ann. Mat. Pura Appl., 32 (1951), 295-370.  doi: 10.1007/BF02417964.

[12]

E. Cartan, Sur les nombres de Betti des espaces de groupes clos, C. R. Acad. Sci., Paris, 187 (1928), 196-198. 

[13]

H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, in Colloque de Topologie (Espaces Fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, 57-71.

[14]

H. Cartan, Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, in Colloque de Topologie (Espaces Fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris, 1951, 15-27.

[15]

H. Cartan and S. Eilenberg, Homological Algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999., With an appendix by David A. Buchsbaum, Reprint of the 1956 original.

[16]

P. Cartier, Effacement dans la cohomologie des algèbres de Lie. Semin. Bourbaki, Vol. 3, Exp. No. 116, Soc. Math. France, Paris, 1995, 161-167.

[17]

S. C. Chang, On Jacobi identity, Acta Math. Sinica, 4 (1954), 365-379. 

[18]

J. Duskin, Simplicial Methods and the Interpretation of "Triple" Cohomology, Mem. Amer. Math. Soc., 3 (1975), v+135 pp. doi: 10.1090/memo/0163.

[19]

S. Eilenberg and S. MacLane, Group extensions and homology, Ann. of Math. (2), 43 (1942), 757-831. doi: 10.2307/1968966.

[20]

S. Eilenberg and S. MacLane, Cohomology theory in abstract groups. I, Ann. of Math. (2), 48 (1947), 51-78. doi: 10.2307/1969215.

[21]

S. Eilenberg and S. MacLane, Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel, Ann. of Math. (2), 48 (1947), 326-341. doi: 10.2307/1969174.

[22]

S. Eilenberg and S. MacLane, Cohomology and Galois theory. I. Normality of algebras and Teichmüller's cocycle, Trans. Amer. Math. Soc., 64 (1948), 1-20.  doi: 10.2307/1990556.

[23]

T. M. Fiore, Pseudo algebras and pseudo double categories, J. Homotopy Relat. Struct., 2 (2007), 119-170. 

[24]

A. Fröhlich and C. T. C. Wall, Equivariant Brauer groups, in Quadratic forms and their Applications (Dublin, 1999), vol. 272 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2000, 57-71. doi: 10.1090/conm/272/04397.

[25]

M. Gerstenhaber, A uniform cohomology theory for algebras, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 626-629.  doi: 10.1073/pnas.51.4.626.

[26]

M. Gerstenhaber, On the deformation of rings and algebras. II, Ann. of Math. (2), 84 (1966), 1-19. doi: 10.2307/1970528.

[27]

S. I. Goldberg, Extensions of Lie algebras and the third cohomology group, Canad. J. Math., 5 (1953), 470-476.  doi: 10.4153/cjm-1953-054-8.

[28]

B. Grossman, The meaning of the third cocycle in the group cohomology of nonabelian gauge theories, Phys. Lett. B, 160 (1985), 94-100.  doi: 10.1016/0370-2693(85)91472-8.

[29]

M. Hall, A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc., 1 (1950), 575-581.  doi: 10.1090/S0002-9939-1950-0038336-7.

[30]

P. Hall, A contribution to the theory of groups of prime-power order, Proc. Lond. Math. Soc., 36 (1934), 29-95.  doi: 10.1112/plms/s2-36.1.29.

[31]

K. HauserF. HerrlichM. KneserH. OpolkaN. Schappacher and E. Scholz, Oswald Teichmüller-leben und werk, Jahresber. Deutsch. Math.-Verein., 94 (1992), 1-39. 

[32]

J.-C. Herz, Pseudo-algèbres de Lie. I, C. R. Acad. Sci. Paris, 236 (1953), 1935-1937. 

[33]

J.-C. Herz, Pseudo-algèbres de Lie. II, C. R. Acad. Sci. Paris, 236 (1953), 2289-2291. 

[34]

P. J. Higgins and K. C. H. Mackenzie, Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures, Math. Proc. Cambridge Philos. Soc., 114 (1993), 471-488.  doi: 10.1017/S0305004100071760.

[35]

P. Hilton, Reminiscences and reflections of a codebreaker, in, Coding Theory and Cryptography (Annapolis, MD, 1998), Springer, Berlin, 2000, 1-8.

[36]

P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc., 30 (1955), 154-172.  doi: 10.1112/jlms/s1-30.2.154.

[37]

P. J. Hilton, Note on quasi-Lie rings, Fund. Math., 43 (1956), 230-237.  doi: 10.4064/fm-43-2-230-237.

[38]

G. Hochschild, Lie algebra kernels and cohomology, Amer. J. Math., 76 (1954), 698-716.  doi: 10.2307/2372712.

[39]

G. Hochschild, Simple algebras with purely inseparable splitting fields of exponent $1$, Trans. Amer. Math. Soc., 79 (1955), 477-489.  doi: 10.2307/1993043.

[40]

G. Hochschild and J.-P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc., 74 (1953), 110-134.  doi: 10.1090/S0002-9947-1953-0052438-8.

[41]

D. F. Holt, An interpretation of the cohomology groups $H^n(G, M)$, J. Algebra, 60 (1979), 307-318.  doi: 10.1016/0021-8693(79)90084-X.

[42]

J. Huebschmann, Verschränkte n-fache Erweiterungen von Gruppen und Cohomologie, 1977. Diss. Math. ETH Zürich, Nr. 5999. Ref. : Eckmann, B.; Korref. : Stammbach, U.

[43] J. Huebschmann, Cohomology theory of aspherical groups and of small cancellation groups, in Homological Group Theory (Proc. Sympos., Durham, 1977), vol. 36 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 1979. 
[44]

J. Huebschmann, Cohomology theory of aspherical groups and of small cancellation groups, J. Pure Appl. Algebra, 14 (1979), 137-143.  doi: 10.1016/0022-4049(79)90003-3.

[45]

J. Huebschmann, Crossed $n$-fold extensions of groups and cohomology, Comment. Math. Helv., 55 (1980), 302-313.  doi: 10.1007/BF02566688.

[46]

J. Huebschmann, Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence, J. Algebra, 72 (1981), 296-334.  doi: 10.1016/0021-8693(81)90296-9.

[47]

J. Huebschmann, Group extensions, crossed pairs and an eight term exact sequence, J. Reine Angew. Math., 321 (1981), 150-172.  doi: 10.1515/crll.1981.321.150.

[48]

J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math., 408 (1990), 57-113. https://arxiv.org/abs/1303.3903. doi: 10.1515/crll. 1990.408.57.

[49]

J. Huebschmann, On the quantization of Poisson algebras, in Symplectic Geometry and Mathematical Physics (Aix-en-Provence, 1990), vol. 99 of Progr. Math., Birkhäuser Boston, Boston, MA, 1991, 204-233.

[50]

J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Ann. Inst. Fourier (Grenoble), 48(1998), 425-440. https://arxiv.org/abs/dg-ga/9704005.

[51]

J. Huebschmann, Twilled Lie-Rinehart algebras and differential Batalin-Vilkovisky algebras, (1998). https://arxiv.org/abs/math/9811069.

[52]

J. Huebschmann, Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras, in Poisson Geometry (Warsaw, 1998), vol. 51 of Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 2000, 87-102. https://arxiv.org/abs/1303.3414.

[53]

J. Huebschmann, Lie-Rinehart algebras, descent, and quantization, in Galois Theory, Hopf Algebras, and Semiabelian Categories, vol. 43 of Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 2004, 295-316. https://arxiv.org/abs/math/0303016. doi: 10.1090/memo/0814.

[54]

J. Huebschmann, Singular Poisson-Kähler geometry of stratified Kähler spaces and quantization, in Geometry and Quantization, vol. 19 of Trav. Math., Univ. Luxemb., Luxembourg, 2011, 27-63. https://arxiv.org/abs/1103.1584.

[55]

J. Huebschmann, Braids and crossed modules, J. Group Theory, 15 (2012), 57-83. https://arxiv.org/abs/0904.3895. doi: 10.1515/JGT. 2011.095.

[56]

J. Huebschmann, Normality of algebras over commutative rings and the Teichmüller class. I, J. Homotopy Relat. Struct., 13 (2018), 1-70. https://arxiv.org/abs/1512.07264. doi: 10.1007/s40062-017-0173-3.

[57]

J. Huebschmann, Normality of algebras over commutative rings and the Teichmüller class. II, J. Homotopy Relat. Struct., 13 (2018), 71-125. https://arxiv.org/abs/1512.07264. doi: 10.1007/s40062-017-0174-2.

[58]

N. Jacobson, An extension of Galois theory to non-normal and non-separable fields, Amer. J. Math., 66 (1944), 1-29.  doi: 10.2307/2371892.

[59]

V. F. R. Jones, An invariant for group actions, in Algèbres d'opérateurs (Sém., Les Plans-sur-Bex, 1978), vol. 725 of Lecture Notes in Math., Springer, Berlin, 1979, 237-253.

[60]

C. Kassel and J. -L. Loday, Extensions centrales d'algèbres de Lie, Ann. Inst. Fourier (Grenoble), 32 (1982), 119-142 (1983).

[61]

J. -L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque, The Mathematical Heritage of Élie Cartan (Lyon, 1984). (1985), 257-271.

[62]

J. -L. Koszul, Lectures on Fibre Bundles and Differential Geometry, vol. 20 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-02503-1.

[63]

A. G. Kurosch, Gruppentheorie. I, Aus dem Russischen übersetzt von Reinhard Strecker und Heidemarie Strecker. Herausgegeben von Reinhard Strecker. Zweite, überarbeitete und erweiterte Auflage. Mathematische Lehrbücher und Monographien, I. Abteilung, Mathematische Lehrbücher, Band III/I, Akademie-Verlag, Berlin, 1970.

[64]

A. G. Kurosch, Gruppentheorie. II, Akademie-Verlag, Berlin, 1972., Aus dem Russischen übersetzt von Reinhard Strecker und Heidemarie Strecker. Herausgegeben von Reinhard Strecker, Zweite, überarbeitete und erweiterte Auflage, Mathematische Lehrbücher und Monographien, I. Abteilung, Mathematische Lehrbücher, Band III/II.

[65]

C. R. Leedham-Green and S. McKay, Baer-invariants, isologism, varietal laws and homology, Acta Math., 137 (1976), 99-150.  doi: 10.1007/BF02392415.

[66]

J.-L. Loday, Cohomologie et groupe de Steinberg relatifs, J. Algebra, 54 (1978), 178-202.  doi: 10.1016/0021-8693(78)90025-X.

[67]

R. C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2), 52 (1950), 650-665. doi: 10.2307/1969440.

[68] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, vol. 124 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511661839.
[69]

K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147.  doi: 10.1112/blms/27.2.97.

[70] K. C. H. Mackenzie, General Theory of Lie groupoids and Lie Algebroids, vol. 213 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9781107325883.
[71]

S. MacLane, Cohomology theory in abstract groups. III. Operator homomorphisms of kernels, Ann. of Math. (2), 50 (1949), 736-761. doi: 10.2307/1969561.

[72]

S. MacLane, Homology. Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin, first ed., 1967.

[73]

S. MacLane, Historical note, J. Algebra, 60 (1979), 319-320. Appendix to [41]. doi: 10.1016/0021-8693(79)90085-1.

[74]

S. MacLane and J. H. C. Whitehead, On the $3$-type of a complex, Proc. Nat. Acad. Sci. U. S. A., 36 (1950), 41-48.  doi: 10.1073/pnas.36.1.41.

[75]

W. Magnus, Über Beziehungen zwischen höheren Kommutatoren, J. Reine Angew. Math., 177 (1937), 105-115.  doi: 10.1515/crll.1937.177.105.

[76]

W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Interscience Publishers [John Wiley & Sons, Inc. ], New York-London-Sydney, 1966.

[77] W. Metzler, Äquivalenzklassen von Gruppenbeschreibungen, Identitäten und einfacher Homotopietyp in niederen Dimensionen, in Homological Group Theory (Proc. Sympos., Durham, 1977), vol. 36 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 1979. 
[78]

M. Mori, On the three-dimensional cohomology group of Lie algebras, J. Math. Soc. Japan, 5 (1953), 171-183.  doi: 10.2969/jmsj/00520171.

[79]

M. Nakaoka and H. Toda, On Jacobi identity for Whitehead products, J. Inst. Polytech. Osaka City Univ. Ser. A, 5 (1954), 1-13. 

[80]

K.-H. Neeb, Non-abelian extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), 57 (2007), 209-271. 

[81]

W. Nichols and B. Weisfeiler, Differential formal groups of J. F. Ritt, Amer. J. Math., 104 (1982), 943-1003.  doi: 10.2307/2374080.

[82]

E. Noether, Hyperkomplexe Größen und Darstellungstheorie, Math. Z., 30 (1929), 641-692.  doi: 10.1007/BF01187794.

[83]

E. Noether, Hyperkomplexe Größen und Darstellungstheorie in arithmetischer Auffassung, Atti Congresso Bologna (1928), 2 (1930), 71-73. 

[84]

K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math., 76 (1954), 33-65.  doi: 10.2307/2372398.

[85]

K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118 (1990), 129-146. 

[86]

R. S. Palais, The cohomology of Lie rings, in Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 130-137.

[87]

C. D. Papakyriakopoulos, Attaching $2$-dimensional cells to a complex, Ann. of Math. (2), 78 (1963), 205-222. doi: 10.2307/1970340.

[88]

R. Peiffer, Über Identitäten zwischen Relationen, Math. Ann., 121 (1949), 67-99.  doi: 10.1007/BF01329617.

[89]

J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245-A248.

[90]

J. G. Ratcliffe, The theory of crossed modules with applications to cohomology of groups and combinatorial homotopy theory, ProQuest LLC, Ann Arbor, MI, 1977., Thesis (Ph. D. )-University of Michigan.

[91]

K. Reidemeister, Knoten und Gruppen, Abh. Math. Semin. Univ. Hamb., 5 (1926), 7-23.  doi: 10.1007/BF02952506.

[92]

K. Reidemeister, Knotentheorie, vol. 1, Springer-Verlag, Berlin, 1932.

[93]

K. Reidemeister, Homotopiegruppen von Komplexen, Abh. Math. Semin. Univ. Hamb., 10 (1934), 211-215.  doi: 10.1007/BF02940675.

[94]

K. Reidemeister, Topologie der Polyeder und kombinatorische Topologie der Komplexe. (Math. u. ihre Anwendungen in Monogr. u. Lehrbüchern. 17). Leipzig: Akad. Verlagsges. mbH IX, 196 S., 69 Fig., 1938.

[95]

K. Reidemeister, Über Identitäten von Relationen, Abh. Math. Sem. Univ. Hamburg, 16 (1949), 114-118.  doi: 10.1007/BF03343521.

[96]

G. S. Rinehart, Differential forms for general commutative algebras, Trans. Amer. Math. Soc., 108 (1963), 195-222.  doi: 10.1090/S0002-9947-1963-0154906-3.

[97]

G. S. Rinehart, Satellites and cohomology, J. Algebra, 12 (1969), 295-329.  doi: 10.1016/0021-8693(69)90032-5.

[98]

J. F. Ritt, Systems of Differential Equations. I. Theory of Ideal, Amer. J. Math., 60 (1938), 535-548.  doi: 10.2307/2371594.

[99]

J. F. Ritt, Associative differential operations, Ann. of Math. (2), 51 (1950), 756-765. doi: 10.2307/1969379.

[100]

J. F. Ritt, Differential Algebra, American Mathematical Society Colloquium Publications, Vol. XXXIII, American Mathematical Society, New York, N. Y., 1950.

[101]

J. F. Ritt, Differential groups and formal Lie theory for an infinite number of parameters, Ann. of Math. (2), 52 (1950), 708-726. doi: 10.2307/1969444.

[102]

H. Samelson, A connection between the Whitehead and the Pontryagin product, Amer. J. Math., 75 (1953), 744-752.  doi: 10.2307/2372549.

[103]

O. Schreier, Über die Erweiterung von Gruppen I, Monatsh. Math. Phys., 34 (1926), 165-180.  doi: 10.1007/BF01694897.

[104]

A. J. Sieradski, Combinatorial isomorphisms and combinatorial homotopy equivalences, J. Pure Appl. Algebra, 7 (1976), 59-95.  doi: 10.1016/0022-4049(76)90067-0.

[105]

P. A. Smith, The complex of a group relative to a set of generators. I, Ann. of Math. (2), 54 (1951), 371-402. doi: 10.2307/1969538.

[106]

J. D. Stasheff, Continuous cohomology of groups and classifying spaces, Bull. Amer. Math. Soc., 84 (1978), 513-530.  doi: 10.1090/S0002-9904-1978-14488-7.

[107]

H. Suzuki, A product in homotopy theory, Tohoku Math. J., 6 (1954), 78-88.  doi: 10.2748/tmj/1178245238.

[108]

O. Teichmüller, Über die sogenannte nichtkommutative Galoissche Theorie und die Relation $\xi_\lambda, \mu, \nu\xi_\lambda, \mu\nu, \pi\xi^\lambda_\mu, \nu, \pi = \xi_\lambda, \mu, \nu \pi\xi_\lambda, \mu, \nu, \pi$, Deutsche Math., 5 (1940), 138-149. 

[109]

A. M. Turing, The extensions of a group, Compositio Math., 5 (1938), 357-367. 

[110] H. Uehara and W. S. Massey, The Jacobi identity for Whitehead products, in Algebraic Geometry and Topology. A symposium in Honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957. 
[111]

G. W. Whitehead, On mappings into group-like spaces, Comment. Math. Helv., 28 (1954), 320-328.  doi: 10.1007/BF02566938.

[112]

J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2), 42 (1941), 409-428. doi: 10.2307/1968907.

[113]

J. H. C. Whitehead, Note on a previous paper entitled "On adding relations to homotopy groups. ", Ann. of Math. (2), 47 (1946), 806-810. doi: 10.2307/1969237.

[114]

J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc., 55 (1949), 453-496.  doi: 10.1090/S0002-9904-1949-09213-3.

[115]

E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math., 177 (1937), 152-160. doi: 10.1515/crll. 1937.177.152.

[116]

Y. C. Wu, $H^{3}(G, A)$ and obstructions of group extensions, J. Pure Appl. Algebra, 12 (1978), 93-100.  doi: 10.1016/0022-4049(78)90023-3.

[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]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[10]

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

[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]

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

[13]

Naoki Chigira, Nobuo Iiyori and Hiroyoshi Yamaki. Nonabelian Sylow subgroups of finite groups of even order. Electronic Research Announcements, 1998, 4: 88-90.

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295

[19]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[20]

Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras. Journal of Modern Dynamics, 2009, 3 (1) : 121-158. doi: 10.3934/jmd.2009.3.121

2021 Impact Factor: 0.737

Metrics

  • PDF downloads (321)
  • HTML views (357)
  • Cited by (0)

Other articles
by authors

[Back to Top]