Lie superalgebra

Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry.

) that is anticommutative and has a graded Jacobi identity also has a

grading; this is the "rolling up" of the algebra into odd and even parts.

This rolling-up is not normally referred to as "super".

Thus, supergraded Lie superalgebras carry a pair of

These two gradations must be compatible, and there is often disagreement as to how they should be regarded.

[1] Formally, a Lie superalgebra is a nonassociative Z2-graded algebra, or superalgebra, over a commutative ring (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading): Super skew-symmetry: The super Jacobi identity:[2] where x, y, and z are pure in the Z2-grading.

When the ground ring is the integers or the Lie superalgebra is a free module, these conditions are equivalent to the condition that the Poincaré–Birkhoff–Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold).

Lie superalgebras show up in physics in several different ways.

This corresponds to a bracket that has a grading of zero: This is not always the case; for example, in BRST supersymmetry and in the Batalin–Vilkovisky formalism, it is the other way around, which corresponds to the bracket of having a grading of -1: This distinction becomes particularly relevant when an algebra has not one, but two graded associative products.

In addition to the Lie bracket, there may also be an "ordinary" product, thus giving rise to the Poisson superalgebra and the Gerstenhaber algebra.

Such gradings are also observed in deformation theory.

By inspecting the Jacobi identity, one sees that there are eight cases depending on whether arguments are even or odd.

These fall into four classes, indexed by the number of odd elements:[3] Thus the even subalgebra

such that, Conditions (1)–(3) are linear and can all be understood in terms of ordinary Lie algebras.

Condition (4) is nonlinear, and is the most difficult one to verify when constructing a Lie superalgebra starting from an ordinary Lie algebra (

Its universal enveloping algebra would be an ordinary *-algebra.

[4] With the Lie bracket per above, the space is denoted

The Whitehead product on homotopy groups gives many examples of Lie superalgebras over the integers.

The super-Poincaré algebra generates the isometries of flat superspace.

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

consisting of matrices with super trace zero.

Consider an even, non-degenerate, supersymmetric bilinear form

consisting of matrices that leave this form invariant:

There is a family of (9∣8)-dimensional Lie superalgebras depending on a parameter

There are also two so-called strange series called

For the Cartan type of simple Lie superalgebras, the odd part is no longer completely reducible under the action of the even part.

The classification consists of the 10 series W(m, n), S(m, n) ((m, n) ≠ (1, 1)), H(2m, n), K(2m + 1, n), HO(m, m) (m ≥ 2), SHO(m, m) (m ≥ 3), KO(m, m + 1), SKO(m, m + 1; β) (m ≥ 2), SHO ~ (2m, 2m), SKO ~ (2m + 1, 2m + 3) and the five exceptional algebras: The last two are particularly interesting (according to Kac) because they have the standard model gauge group SU(3)×SU(2)×U(1) as their zero level algebra.

Infinite-dimensional (affine) Lie superalgebras are important symmetries in superstring theory.

[7] In category theory, a Lie superalgebra can be defined as a nonassociative superalgebra whose product satisfies where σ is the cyclic permutation braiding