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

Abstract

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.

21 pages. v2: published version

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Cited by in corpus (24)

• n-ary algebras: a review with applications
• Multiple Membranes in M-theory
• The tensor hierarchy algebra
• Tensor hierarchies, Borcherds algebras and E11
• Higgsing M2 to D2 with gravity: N=6 chiral supergravity from topologically gauged ABJM theory
• Three-dimensional topologically gauged N=6 ABJM type theories
• Symplectic Three-Algebra Unifying N=5,6 Superconformal Chern-Simons-Matter Theories
• Exceptional geometry and Borcherds superalgebras
• Super $w_{\infty }$ 3-algebra
• Monopoles, three-algebras and ABJM theories with $\mathbb{N}=5,6,8$ supersymmetry
• Tensor hierarchy algebras and extended geometry I: Construction of the algebra
• Simplicity in the Faulkner construction
• Superspace Formulation in a Three-Algebra Approach to D=3, N=4,5 Superconformal Chern-Simons Matter Theories
• Infinity-enhancing of Leibniz algebras
• N=5 three-algebras and 5-graded Lie superalgebras
• Aspects of topologically gauged M2-branes with six supersymmetries: towards a "sequential AdS/CFT"?
• Superalgebra Realization of the 3-algebras in N=6, 8 Chern-Simons-matter Theories
• Lie $3-$algebra and super-affinization of split-octonions
• Classification of linearly compact simple N=6 3-algebras
• Algebraic vs physical N=6 3-algebras
• k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems
• Nonlinear realisations of Lie superalgebras
• A class of hermitian generalized Jordan triple systems and Chern-Simons gauge theory
• Metric 3-Leibniz algebras and M2-branes