Built on

N. Jacobson, “General representation theory of Jordan algebras,” Trans. Amer. Math. Soc
1951
Earlier work this paper cites.
W. G. Lister, “A structure theory of Lie triple systems,” Trans. Am. Math. Soc
1952
Earlier work this paper cites.
K. Yamaguti, “On algebras of totally geodesic spaces (Lie triple systems),” J. Sci. Hiroshima Univ. Ser. A
1958
Earlier work this paper cites.
J. R. Faulkner, “On the geometry of inner ideals,” J. Algebra
1973
Earlier work this paper cites.
Y. Nambu, “Generalized Hamiltonian dynamics,” Phys. Rev
1973
Earlier work this paper cites.
J. R. Faulkner and J. C. Ferrar, “Simple anti-Jordan pairs,” Comm. Algebra
1980
Earlier work this paper cites.
V. Filippov, “ n n -Lie algebras,” Sibirsk. Mat. Zh
1985
Earlier work this paper cites.
S. Okubo, “Construction of Lie superalgebras from triple product systems,” AIP Conf. Proc
2003
Earlier work this paper cites.

Similar

N. Kamiya and S. Okubo, “Construction of Lie superalgebras D ⁡ ( 2 , 1 , α ) , G ⁡ ( 3 ) D(2,1;\alpha),\ G(3) and F ⁡ ( 4 ) F(4) from some triple systems,” Proc. Edinb. Math. Soc. (2)
2003
Cited alongside, same era.
J. H. Schwarz, “Superconformal Chern-Simons theories,” JHEP
2004
Cited alongside, same era.
J. M. Figueroa-O’Farrill and G. Papadopoulos, “Plücker-type relations for orthogonal planes,” J. Geom. Phys
2004
Cited alongside, same era.
J. M. Figueroa-O’Farrill, P. Meessen, and S. Philip, “Supersymmetry and homogeneity of M-theory backgrounds,” Class. Quant. Grav
2005
Cited alongside, same era.
J. Bagger and N. Lambert, “Modeling multiple M2’s,” Phys. Rev
2007
Cited alongside, same era.
D. Gaiotto and X. Yin, “Notes on superconformal Chern-Simons-matter theories,” JHEP

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2008

On the Lie-algebraic origin of metric 3-algebras

de Medeiros, Paul, Figueroa-O'Farrill, José, Méndez-Escobar, Elena et al.

Understand

Since the pioneering work of Bagger-Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern-Simons theories whose main ingredient is a metric 3-algebra.

On the other hand, many of these theories have been shown to allow for a reformulation in terms of standard gauge theory coupled to matter, where the 3-algebra does not appear explicitly.
In this paper we reconcile these two sets of results by pointing out the Lie-algebraic origin of some metric 3-algebras, including those which have already appeared in three-dimensional superconformal Chern-Simons theories.
More precisely, we show that the real 3-algebras of Cherkis-Saemann, which include the metric Lie 3-algebras as a special case, and the hermitian 3-algebras of Bagger-Lambert can be constructed from pairs consisting of a metric real Lie algebra and a faithful (real or complex, respectively) unitary representation.

Built on

N. Jacobson, “General representation theory of Jordan algebras,” Trans. Amer. Math. Soc
1951
Earlier work this paper cites.
W. G. Lister, “A structure theory of Lie triple systems,” Trans. Am. Math. Soc
1952
Earlier work this paper cites.
K. Yamaguti, “On algebras of totally geodesic spaces (Lie triple systems),” J. Sci. Hiroshima Univ. Ser. A
1958
Earlier work this paper cites.
J. R. Faulkner, “On the geometry of inner ideals,” J. Algebra
1973
Earlier work this paper cites.
Y. Nambu, “Generalized Hamiltonian dynamics,” Phys. Rev
1973
Earlier work this paper cites.
J. R. Faulkner and J. C. Ferrar, “Simple anti-Jordan pairs,” Comm. Algebra
1980
Earlier work this paper cites.
V. Filippov, “ n n -Lie algebras,” Sibirsk. Mat. Zh
1985
Earlier work this paper cites.
S. Okubo, “Construction of Lie superalgebras from triple product systems,” AIP Conf. Proc
2003
Earlier work this paper cites.

Similar

N. Kamiya and S. Okubo, “Construction of Lie superalgebras D ⁡ ( 2 , 1 , α ) , G ⁡ ( 3 ) D(2,1;\alpha),\ G(3) and F ⁡ ( 4 ) F(4) from some triple systems,” Proc. Edinb. Math. Soc. (2)
2003
Cited alongside, same era.
J. H. Schwarz, “Superconformal Chern-Simons theories,” JHEP
2004
Cited alongside, same era.
J. M. Figueroa-O’Farrill and G. Papadopoulos, “Plücker-type relations for orthogonal planes,” J. Geom. Phys
2004
Cited alongside, same era.
J. M. Figueroa-O’Farrill, P. Meessen, and S. Philip, “Supersymmetry and homogeneity of M-theory backgrounds,” Class. Quant. Grav
2005
Cited alongside, same era.
J. Bagger and N. Lambert, “Modeling multiple M2’s,” Phys. Rev
2007
Cited alongside, same era.
D. Gaiotto and X. Yin, “Notes on superconformal Chern-Simons-matter theories,” JHEP