×

The Tits-Kantor-Koecher construction for Jordan dialgebras. (English) Zbl 1272.17032

The authors study a noncommutative generalization of Jordan algebras called Jordan dialgebras; these algebras are related with Jordan algebras in the same way as Leibniz algebras are related to Lie algebras. The authors present an analogue of the Tits-Kantor-Koecher (TKK) construction for Jordan dialgebras that provides an embedding of such an algebra into Leibniz algebra. Some “classical” results on solvable and nilpotent Jordan dialgebras are given as well.

MSC:

17C99 Jordan algebras (algebras, triples and pairs)
17A32 Leibniz algebras
17B69 Vertex operators; vertex operator algebras and related structures
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albert A. A., Ann. of Math. 2 pp 546– (1947) · Zbl 0029.01003 · doi:10.2307/1969128
[2] Bakalov B., Adv. Math. 162 pp 1– (2001) · Zbl 1001.16021 · doi:10.1006/aima.2001.1993
[3] Beilinson A. A., Chiral Algebras (2004) · Zbl 1138.17300
[4] Bremner M. R., Comm. Algebra
[5] Ginzburg V., Duke Math. J. 76 pp 203– (1994) · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4
[6] Faulkner J. R., Geom. Dedicata 30 pp 125– (1989)
[7] Jacobson N., Structure and Representations of Jordan Algebras (1968) · Zbl 0218.17010
[8] Kac V. G., Vertex Algebras for Beginners (1996) · Zbl 0861.17017
[9] Kac V. G., Proc. of XII International Congress in Mathematical Physics (ICMP’97) pp 80– (1999)
[10] Kolesnikov P. S., Comm. Algebra 34 pp 1965– (2006) · Zbl 1144.17020 · doi:10.1080/00927870500542945
[11] Kolesnikov P. S., Siberian Math. J. 49 pp 257– (2008) · doi:10.1007/s11202-008-0026-8
[12] Kolesnikov P. S., SIGMA (2009)
[13] Lambek , J. ( 1969 ). Deductive systems and categories II.Standard Constructions and Closed Categories.In: Category Theory, Homology Theory and Their Applications. Proceedings of the Conference Held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July 19, 1968 Category Theory, Homology Theory and Their Applications. Proceedings of the Conference Held at the Seattle Research Center of the Battelle Memorial Institute, June 24–July 19, 1968. Hilton, P. J., ed. Series: Lecture Notes Math., 86. Berlin: Springer-Verlag, pp. 76–122 .
[14] Liu D., J. Algebra 283 pp 199– (2005) · Zbl 1071.17001 · doi:10.1016/j.jalgebra.2004.08.005
[15] Loday J.-L., Enseign. Math. 39 pp 269– (1993)
[16] Loday J.-L., Cyclic cohomology and noncommutative geometry pp 91– (1997)
[17] Loday J.-L., Dialgebras and Related Operads pp 7– (2001) · doi:10.1007/3-540-45328-8_2
[18] Loday J.-L., Math. Ann. 296 pp 139– (1993) · Zbl 0821.17022 · doi:10.1007/BF01445099
[19] Roitman M., J. Algebra 217 pp 496– (1999) · Zbl 0951.17013 · doi:10.1006/jabr.1998.7834
[20] Pozhidaev A. P., Groups, Rings and Group Rings pp 245– (2009) · Zbl 1224.17003
[21] Velasquez R., Comm. Algebra 36 pp 1580– (2008) · Zbl 1188.17021 · doi:10.1080/00927870701865996
[22] Zel’manov E. I., Trudy Inst. Mat. 16 pp 37– (1989)
[23] Zelmanov E. I., Proceedings of Intern. Conf. on Combinatorial and Computational Algebra (May 24–29, 1999, Hong Kong) (2000)
[24] Zhevlakov K. A., Rings that are Nearly Associative (1982) · Zbl 0487.17001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.