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Background

Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) and Q(n), and the basic superalgebras:

  • the A series $sl(m,n)$ ($psl(m,m)$ for $m=n$)

  • the B, C, D series $osp(m,n)$

  • a family D(2,1;$\alpha$) of deformations of D(2,1)

  • the exceptional Lie algebras F(4) and G(3)

Basic Lie superalgebras have roots, Cartan matrices and Dynkin diagrams, and their Dynkin diagrams look quite similar to usual Dynkin diagrams of simple Lie algebras. A noticeable difference is that the Dynkin diagram is not unique anymore (so that diagrams of type C and D for example describe the same Lie superalgebras).

Question

Why is the E series missing? More precisely, are there known Lie superalgebras that are defined similarly to the Lie algebras $E_6$, $E_7$, $E_8$ (e.g. via octonions as in the classical case) and why are they not simple or finite dimensional?

Remarks

A natural place to look for Lie superalgebras of type E is the class of Kac-Moody superalgebras (KMSA). The simplest infinite dimensional Kac-Moody superalgebras are the affine ones. Their construction is similar to the classical one, and there are no examples that "look like" algebras of type E. Expanding the search to KMSA of finite growth only gives a few exceptional families, that again are not similar to algebras of type E (see arXiv:0810.2637).

The next simplest class of infinite dimensional KMSA is that of hyperbolic KMSA, i.e. those such that removing a root from (any of) its Dynkin diagram(s) yields a Dynkin diagram of a finite dimensional or affine KMSA. There have been at least three attempts at classification of hyperbolic KMSA: 1, 2, 3. Some diagrams of type E indeed appear, however they are just listed, and the resulting KMSA are only described by their Cartan matrix. Have these algebras been described in more detail somewhere?

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1 Answer

I think an useful reference will be http://www.sciencedirect.com/science/article/pii/0001870877900172 (V.G Kac, Lie superalgebras).

Exceptional Lie superalgebras G(3) and F(4) can indeed be expressed in terms of octonions: http://jmp.aip.org/resource/1/jmapaq/v24/i8/p1986_s1 (A. Sudbery, Octonionic description of exceptional Lie superalgebras). See also http://link.springer.com/article/10.1007%2Fs00014-003-0790-9 (A. Elduque, Quaternions, octonions and the forms of the exceptional simple classical Lie superalgebras).

infinite dimensional case is considered in http://xxx.lanl.gov/abs/math.QA/9912235 (V.G Kac, Classification of infinite-dimensional simple groups of supersymmetries and quantum field theory).

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