# Three-algebras, triple systems and 3-graded Lie superalgebras

@article{Palmkvist2009ThreealgebrasTS, title={Three-algebras, triple systems and 3-graded Lie superalgebras}, author={Jakob Palmkvist}, journal={Journal of Physics A}, year={2009}, volume={43}, pages={015205} }

The three-algebras used by Bagger and Lambert in N = 6 theories of ABJM type are in one-to-one correspondence with a certain type of Lie superalgebras. We show that the description of three-algebras as generalized Jordan triple systems naturally leads to this correspondence. Furthermore, we show that simple three-algebras correspond to simple Lie superalgebras, and vice versa. This gives a classification of simple three-algebras from the well-known classification of simple Lie superalgebras.

#### 36 Citations

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We discuss a generalization of N=6 three-algebras to N=5 three-algebras in connection to anti-Lie triple systems and basic Lie superalgebras of type II. We then show that the structure constants… Expand

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Two types of higher order Lie l-ple systems are introduced in this paper. They are defined by brackets with l > 3 arguments satisfying certain conditions, and generalize the well known Lie triple… Expand

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We investigate the super high-order Virasoro-Witt 3-algebra. By applying the appropriate scaling limits on the generators, we obtain the super w∞ 3-algebra which satisfies the generalized fundamental… Expand

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- 2011

N ≤ 8 3-algebras have recently appeared in N-supersymmetric 3-dimensional Chern–Simons gauge theories. In our previous paper we classified linearly compact simple N = 8 n-algebras for any n ≥ 3. In… Expand

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We write the Lagrangian of the general N=5 three-dimensional superconformal Chern-Simons theory, based on a basic Lie superalgebra, in terms of our recently introduced N=5 three-algebras. These… Expand

ON THE STRUCTURE OF GRADED LIE TRIPLE SYSTEMS

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- 2016

We study the structure of an arbitrary graded Lie triple sys- tem T with restrictions neither on the dimension nor the base field. We show that T is of the form T = U + P j Ij with U a linear… Expand

A class of Hermitian generalized Jordan triple systems and Chern-Simons gauge theory

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We find a class of Hermitian generalized Jordan triple systems (HGJTSs) and Hermitian (ϵ, δ)-Freudenthal–Kantor triple systems (HFKTSs). We apply one of the most simple HGJTSs which we find to a… Expand

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We revisit the Faulkner construction of metric 3-Leibniz algebras admitting an embedding Lie (super)algebra. In the case of positive-definite signature, we relate the various notions of simplicity:… Expand

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