Lie Superalgebra - Classification

Classification

The simple complex finite dimensional Lie superalgebras were classified by Victor Kac.

The basic classical compact Lie superalgebras (that are not Lie algebras) are:

SU(m/n) These are the superunitary Lie algebras which have invariants:

This gives two orthosymplectic (see below) invariants if we take the m z variables and n w variables to be non-commuative and we take the real and imaginary parts. Therefore we have

SU(n/n)/U(1) A special case of the superunitary Lie algebras where we remove one U(1) generator to make the algebra simple.

OSp(m/2n) These are the Orthosymplectic groups. They have invariants given by:

for m commutative variables (x) and n pairs of anti-commuative variables (y,z). They are important symmetries in supergravity theories.

D(2/1;) This is a set of superalgebras paramaterised by the variable . It has dimension 17 and is a sub-algebra of OSp(9|8). The even part of the group is O(3)xO(3)xO(3). So the invariants are:

$A_{\{1} A_2 A_{3\}} + B_{\{1} B_2 B_{3\}} + C_{\{1} C_2 C_{3\}} + A_\mu \Gamma^{\alpha \alpha'}_\mu \psi\psi + B_\mu \Gamma^{\beta \beta'}_\mu \psi\psi + C^{\gamma \gamma'}_\mu \Gamma_\mu \psi\psi$

for particular constants .

F(4) This exceptional Lie superalgebra has dimension 40 and is a sub-algebra of OSp(24|16). The even part of the group is O(3)xSO(7) so three invariants are:

This group is related to the octonions by considering the 16 component spinors as two component octonion spinors and the gamma matrices acting on the upper indices as unit octonions. We then have where f is the structure constants of octonion multiplication.

G(3) This exceptional Lie superalgebra has dimension 31 and is a sub-algebra of OSp(17|14). The even part of the group is O(3)xG2. The invariants are similar to the above (it being a subalgebra of the F(4)?) so the first invariant is:

There are also two so-called strange series called p(n) and q(n).

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