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A compendium of Lie structures on tensor products
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 40-81 Cet article a éte moissonné depuis la source Math-Net.Ru

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We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads. 
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P. Zusmanovich. A compendium of Lie structures on tensor products. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 40-81. http://geodesic.mathdoc.fr/item/ZNSL_2013_414_a3/

[1] C. Bai, D. Meng, “The classification of Novikov algebras in low dimensions”, J. Phys. A, 34 (2001), 1581–1594 | DOI | MR | Zbl

[2] C. Bai, D. Meng, L. He, “On fermionic Novikov algebras”, J. Phys. A, 35 (2002), 10053–10063 | DOI | MR | Zbl

[3] Soviet Math. Dokl., 32 (1985), 228–231 | MR | Zbl

[4] G. Benkart, E. Neher, “The centroid of extended affine and root graded Lie algebras”, J. Pure Appl. Algebra, 205 (2006), 117–145 ; arXiv: math/0502561 | DOI | MR | Zbl

[5] G. Benkart, J. M. Osborn, “Flexible Lie-admissible algebras”, J. Algebra, 71 (1981), 11–31 | DOI | MR | Zbl

[6] F. A. Berezin, Introduction to Superanalysis, with an appendix, Seminar on Supersymmetry, 1$\frac12$, 2nd revised ed., ed. D. Leites, MCCME (to appear) (in Russian) | MR

[7] S. Bouarroudj, A. Lebedev, D. Leites, I. Shchepochkina, Deforms of Lie algebras in characteristic 2: semi-trivial for Jurman algebras, non-trivial for Kaplansky algebras, arXiv: 1301.2781v1

[8] Hermann, 1972 | MR | Zbl

[9] J. L. Cathelineau, “Homologie de degré trois d'algèbres de Lie simple déployées étendues à une algèbre commutative”, Enseign. Math., 33 (1987), 159–173 ; “Correction”, Algebra i Logika, 28:2 (1989), 241 (in Russian) | MR | Zbl | MR | Zbl

[10] Algebra and Logic, 24:1 (1985), 1–7 ; “Correction”, Algebra and Logic, 28:2 (1989), 162 | DOI | DOI | MR | MR | Zbl | Zbl

[11] Math. USSR Sbornik, 66:2 (1990), 461–473 | DOI | MR | Zbl | Zbl

[12] A. S. Dzhumadil'daev, “Codimension growth and non-Koszulity of Novikov operad”, Comm. Algebra, 39 (2011), 2943–2952 ; loosely corresponds to: arXiv: + 0902.31870902.3771 | DOI | MR | Zbl

[13] R. Farnsteiner, H. Strade, “Shapiro's lemma and its consequences in the cohomology theory of modular Lie algebras”, Math. Z., 206 (1991), 153–168 | DOI | MR | Zbl

[14] J. Feldvoss, H. Strade, “Restricted Lie algebras with bounded cohomology and related classes of algebras”, Manuscr. Math., 74 (1992), 47–67 | DOI | MR | Zbl

[15] Funct. Anal. Appl., 13 (1979), 248–262 | DOI | MR | Zbl

[16] Russ. Math. Surv., 23:2 (1968), 1–58 | DOI | MR | Zbl

[17] V. Ginzburg, M. Kapranov, “Koszul duality for operads”, Duke Math. J., 76 (1994), 203–272 ; “Erratum”, Duke Math. J., 80 (1995), 293 ; 0709.1228 | DOI | MR | Zbl | DOI | MR | Zbl

[18] A. Grishkov, “On simple Lie algebras over a field of characteristic 2”, J. Algebra, 363 (2012), 14–18 | DOI | MR | Zbl

[19] P. Grozman, D. Leites, I. Shchepochkina, “Lie superalgebras of string theories”, Acta Math. Vietnam., 26 (2001), 27–63 ; arXiv: hep-th/9702120 | MR | Zbl

[20] H. Gündoğan, Lie algebras of smooth sections, Diploma Thesis, Technische Univ. Darmstadt, 2007; arXiv: 0803.2800v4

[21] J. Hartwig, D. Larsson, S. Silvestrov, “Deformations of Lie algebras using $\sigma$-derivations”, J. Algebra, 295 (2006), 314–361 ; arXiv: math/0408064 | DOI | MR | Zbl

[22] A. Heller, I. Reiner, “Indecomposable representations”, Illinois J. Math., 5 (1961), 314–323 | MR | Zbl

[23] N. Jacobson, Lie Algebras, Dover, 1979, reprint of Interscience Publ., 1962 | MR

[24] K. Jeong, S.-J. Kang, H. Lee, “Lie-admissible algebras and Kac-Moody algebras”, J. Algebra, 197 (1997), 492–505 | DOI | MR | Zbl

[25] Q. Jin, X. Li, “Hom-Lie algebra structures on semi-simple Lie algebras”, J. Algebra, 319 (2008), 1398–1408 | DOI | MR | Zbl

[26] G. Jurman, “A family of simple Lie algebras in characteristic two”, J. Algebra, 271 (2004), 454–481 | DOI | MR | Zbl

[27] V. G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, 1995 | MR

[28] V. G. Kac, J. W. van de Leur, “On classification of superconformal algebras”, Strings' 88, eds. S. J. Gates, C. R. Preitschopf, W. Siegel, World Scientific, 1989, 77–106 | MR | Zbl

[29] J. Math. Sci., 100 (2000), 1944–2002 | DOI | MR | Zbl

[30] F. Kubo, “Non-commutative Poisson algebra structures on affine Kac-Moody algebras”, J. Pure Appl. Algebra, 126 (1998), 267–286 | DOI | MR | Zbl

[31] F. Kubo, “Compatible algebra structures of Lie algebras”, Ring Theory 2007, eds. H. Marubayashi et al., World Scientific, 2008, 235–239 | MR

[32] P. B. A. Lecomte, C. Roger, “Rigidity of current Lie algebras of complex simple Lie type”, J. London Math. Soc., 37 (1988), 232–240 | DOI | MR | Zbl

[33] J. Soviet Math., 3 (1975), 636–653 | DOI | MR | Zbl

[34] Y. Pei, C. Bai, “Realizations of conformal current-type Lie algebras”, J. Math. Phys., 51 (2010), 052302 | DOI | MR

[35] Y. Pei, C. Bai, “Novikov algebras and Schrödinger-Virasoro Lie algebras”, J. Phys. A, 44 (2011), 045201 | DOI | MR | Zbl

[36] H. Strade, Lie Algebras over Fields of Positive Characteristic, v. I, Structure Theory, de Gruyter, 2004 | MR | Zbl

[37] P. Zusmanovich, “Low-dimensional cohomology of current Lie algebras and analogs of the Riemann tensor for loop manifolds”, Lin. Algebra Appl., 407 (2005), 71–104 ; arXiv: math/0302334 | DOI | MR | Zbl

[38] P. Zusmanovich, “A converse to the Whitehead Theorem”, J. Lie Theory, 18 (2008), 811–815 ; arXiv: 0808.0212 | MR | Zbl

[39] P. Zusmanovich, “Invariants of current Lie algebras extended over commutative algebras without unit”, J. Nonlin. Math. Phys., 17, Suppl. 1, Special issue in memory of F. A. Berezin (2010), 87–102 ; arXiv: 0901.1395 | DOI | MR

[40] P. Zusmanovich, “Non-existence of invariant symmetric forms on generalized Jacobson–Witt algebras revisited”, Comm. Algebra, 39 (2011), 548–554 ; arXiv: 0902.0038 | DOI | MR | Zbl