@article{SIGMA_2015_11_a16,
author = {Igor Buchberger and J\"urgen Fuchs},
title = {On the {Killing} form of {Lie} {Algebras} in {Symmetric} {Ribbon} {Categories}},
journal = {Symmetry, integrability and geometry: methods and applications},
year = {2015},
volume = {11},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a16/}
}
Igor Buchberger; Jürgen Fuchs. On the Killing form of Lie Algebras in Symmetric Ribbon Categories. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a16/
[1] Albuquerque H., Majid S., “Quasialgebra structure of the octonions”, J. Algebra, 220 (1999), 188–224, arXiv: math.QA/9802116 | DOI
[2] Baez J. C., “The octonions”, Bull. Amer. Math. Soc., 39 (2002), 145–205, arXiv: math.RA/0105155 | DOI
[3] Bahturin Y., Fischman D., Montgomery S., “On the generalized Lie structure of associative algebras”, Israel J. Math., 96 (1996), 27–48 | DOI
[4] Bahturin Y. A., Mikhalev A. A., Petrogradsky V. M., Zaicev M. V., Infinite-dimensional Lie superalgebras, de Gruyter Expositions in Mathematics, 7, Walter de Gruyter Co., Berlin, 1992 | DOI
[5] Borcherds R. E., “Quantum vertex algebras”, Taniguchi Conference on Mathematics Nara'98, Adv. Stud. Pure Math., 31, Math. Soc. Japan, Tokyo, 2001, 51–74, arXiv: math.QA/9903038
[6] Caenepeel S., Goyvaerts I., “Monoidal Hom–Hopf algebras”, Comm. Algebra, 39 (2011), 2216–2240, arXiv: 0907.0187 | DOI
[7] Coecke B., Duncan R., “Interacting quantum observables: categorical algebra and diagrammatics”, New J. Phys., 13 (2011), 043016, 85 pp., arXiv: 0906.4725 | DOI
[8] Coecke B., Pavlovic D., Vicary J., “A new description of orthogonal bases”, Math. Structures Comput. Sci., 23 (2013), 555–567, arXiv: 0810.0812 | DOI
[9] Cohen M., Fischman D., Westreich S., “Schur's double centralizer theorem for triangular Hopf algebras”, Proc. Amer. Math. Soc., 122 (1994), 19–29 | DOI
[10] Crane L., Yetter D., “On algebraic structures implicit in topological quantum field theories”, J. Knot Theory Ramifications, 8 (1999), 125–163, arXiv: hep-th/9412025 | DOI
[11] Deligne P., “Catégories tannakiennes”, The Grothendieck Festschrift, v. II, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990, 111–195
[12] Deligne P., Morgan J. W., “Notes on supersymmetry (following Joseph Bernstein)”, Quantum Fields and Strings: a Course for Mathematicians (Princeton, NJ, 1996/1997), v. 1, 2, Amer. Math. Soc., Providence, RI, 1999, 41–97
[13] Dieudonné J., “On semi-simple Lie algebras”, Proc. Amer. Math. Soc., 4 (1953), 931–932 | DOI
[14] Etingof P., Ostrik V., “Finite tensor categories”, Mosc. Math. J., 4 (2004), 627–654, arXiv: math.QA/0301027
[15] Fischman D., Montgomery S., “A Schur double centralizer theorem for cotriangular Hopf algebras and generalized Lie algebras”, J. Algebra, 168 (1994), 594–614 | DOI
[16] Fjelstad J., Fuchs J., Runkel I., Schweigert C., “TFT construction of RCFT correlators. V: Proof of modular invariance and factorisation”, Theory Appl. Categ., 16 (2006), 342–433, arXiv: hep-th/0503194
[17] Freund P. G. O., Kaplansky I., J. Math. Phys., 17 (1976), Simple supersymmetries | DOI
[18] Fuchs J., Runkel I., Schweigert C., “TFT construction of RCFT correlators. I: Partition functions”, Nuclear Phys. B, 646 (2002), 353–497, arXiv: hep-th/0204148 | DOI
[19] Fuchs J., Schweigert C., “Hopf algebras and finite tensor categories in conformal field theory”, Rev. Un. Mat. Argentina, 51 (2010), 43–90, arXiv: 1004.3405
[20] Fuchs J., Stigner C., “On Frobenius algebras in rigid monoidal categories”, Arab. J. Sci. Eng. Sect. C Theme Issues, 33 (2008), 175–191, arXiv: 0901.4886
[21] Goyvaerts I., Vercruysse J., “A note on the categorification of Lie algebras”, Lie Theory and its Applications in Physics, Springer Proceedings in Mathematics Statistics, 36, ed. V. Dobrev, Springer-Verlag, Tokyo, 2013, 541–550, arXiv: 1202.3599 | DOI
[22] Gurevich D. I., “Generalized translation operators on Lie groups”, Soviet J. Contemporary Math. Anal., 18 (1983), 57–70
[23] Gurevich D. I., “Elements of the formal Lie theory and the Poincaré–Birkhoff–Witt theorem for generalized shift operators”, Funct. Anal. Appl., 20 (1986), 315–317 | DOI
[24] Hái P. H., “An embedding theorem for abelian monoidal categories”, Compositio Math., 132 (2002), 27–48, arXiv: math.CT/0004160 | DOI
[25] Happel D., “On the derived category of a finite-dimensional algebra”, Comment. Math. Helv., 62 (1987), 339–389 | DOI
[26] Hartwig J. T., Larsson D., Silvestrov S. D., “Deformations of Lie algebras using $\sigma$-derivations”, J. Algebra, 295 (2006), 314–361, arXiv: math.QA/0408064 | DOI
[27] James G. D., “The modular characters of the Mathieu groups”, J. Algebra, 27 (1973), 57–111 | DOI
[28] Joyal A., Street R., “The geometry of tensor calculus, I”, Adv. Math., 88 (1991), 55–112 | DOI
[29] Joyal A., Street R., Verity D., “Traced monoidal categories”, Math. Proc. Cambridge Philos. Soc., 119 (1996), 447–468 | DOI
[30] Kac V. G., “Lie superalgebras”, Adv. Math., 26 (1977), 8–96 | DOI
[31] Kadison L., Stolin A. A., “Separability and Hopf algebras”, Algebra and its Applications (Athens, OH, 1999), Contemp. Math., 259, Amer. Math. Soc., Providence, RI, 2000, 279–298 | DOI
[32] Kassel C., Quantum groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995 | DOI
[33] Kharchenko V. K., “Connected braided Hopf algebras”, J. Algebra, 307 (2007), 24–48 | DOI
[34] Kissinger A., Pictures of processes: automated graph rewriting for monoidal categories and applications to quantum computing, Ph. D. Thesis, University of Oxford, 2011, arXiv: 1203.0202
[35] Krause H., Krull–Schmidt categories and projective covers, arXiv: 1410.2822
[36] Lyubashenko V., “Modular transformations for tensor categories”, J. Pure Appl. Algebra, 98 (1995), 279–327 | DOI
[37] MacDonald J. L., “Conditions for a universal mapping of algebras to be a monomorphism”, Bull. Amer. Math. Soc., 80 (1974), 888–892 | DOI
[38] Majid S., “Quantum and braided-Lie algebras”, J. Geom. Phys., 13 (1994), 307–356, arXiv: hep-th/9303148 | DOI
[39] Müger M., “Tensor categories: a selective guided tour”, Rev. Un. Mat. Argentina, 51 (2010), 95–163, arXiv: 0804.3587
[40] Pareigis B., “When Hopf algebras are Frobenius algebras”, J. Algebra, 18 (1971), 588–596 | DOI
[41] Pareigis B., “On Lie algebras in braided categories”, Quantum Groups and Quantum Spaces (Warsaw, 1995), Banach Center Publ., 40, Polish Acad. Sci., Warsaw, 1997, 139–158, arXiv: q-alg/9612002
[42] Pareigis B., “On Lie algebras in the category of Yetter–Drinfeld modules”, Appl. Categ. Structures, 6 (1998), 151–175, arXiv: q-alg/9612023 | DOI
[43] Rittenberg V., Wyler D., “Generalized superalgebras”, Nuclear Phys. B, 139 (1978), 189–202 | DOI
[44] Schauenburg P., “A characterization of the Borel-like subalgebras of quantum enveloping algebras”, Comm. Algebra, 24 (1996), 2811–2823 | DOI
[45] Scheunert M., “Generalized Lie algebras”, J. Math. Phys., 20 (1979), 712–720 | DOI
[46] Selinger P., “Dagger compact closed categories and completely positive maps: (extended abstract)”, Electron. Notes Theor. Computer Sci., 170 (2007), 139–163 | DOI
[47] Selinger P., “A survey of graphical languages for monoidal categories”, New Structures for Physics, Lecture Notes in Phys., 813, Springer, Heidelberg, 2011, 289–355, arXiv: 0908.3347 | DOI
[48] Soibelman Ya. S., Meromorphic tensor categories, arXiv: q-alg/9709030
[49] Street R., “Frobenius monads and pseudomonoids”, J. Math. Phys., 45 (2004), 3930–3948 | DOI
[50] Turaev V. G., Virelizier A., On two approaches to 3-dimensional TQFTs, arXiv: 1006.3501
[51] Yamagami S., “Frobenius algebras in tensor categories and bimodule extensions”, Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004, 551–570
[52] Yetter D. N., “Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories”, Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990), Contemp. Math., 134, Amer. Math. Soc., Providence, RI, 1992, 325–349 | DOI
[53] Zhang S., Zhang Y.-Z., “Braided $m$-Lie algebras”, Lett. Math. Phys., 70 (2004), 155–167, arXiv: math.RA/0308095 | DOI