Lie superalgebra

From Academic Kids

In mathematics, a Lie superalgebra is a kind of generalisation of a Lie algebra. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions.

A Lie superalgebra is nonassociative superalgebra which is the graded version of a ordinary Lie algebra. The product map is written as <math>[\cdot,\cdot]<math> instead. Category theoretically, <math>[\cdot,\cdot]\circ (id+\tau_{A,A})=0<math> and <math>[\cdot,\cdot]\circ ([\cdot,\cdot]\otimes id)\circ(id+\sigma+\sigma^2)=0<math> where σ is the cyclic permutation braiding <math>(id\otimes \tau_{A,A})\circ(\tau_{A,A}\otimes id)<math>.

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Liealgebra.png
Image:Liealgebra.png

Translated into more concrete terms, a Lie superalgebra is a Z2-graded algebra over a field of characteristic 0 (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies

<math>[x,y]=-(-1)^{|x| |y|}[y,x]<math>

and

<math>(-1)^{|z| |x|}[x,[y,z]]+(-1)^{|x| |y|}[y,[z,x]]+(-1)^{|y| |z|}[z,[x,y]]=0<math>

where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1).

Lie superalgebras are a natural generalization of normal Lie algebras to include a Z2-grading. Indeed, the above conditions on the superbracket are exactly those on the normal Lie bracket with modifications made for the grading. The last condition is sometimes called the super Jacobi identity.

Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the funny signs disappear, and the superbracket becomes a normal Lie bracket.

One way of thinking about a Lie superalgebra -- it's not the most symmetric way of looking at it -- is to consider its even and odd parts, L0 and L1 separately. Then, L0 is a Lie algebra, L1 is a linear rep of L0, and there exists a symmetric L0-intertwiner <math>\{.,.\}:L_1\otimes L_1\rightarrow L_0<math> such that for all x,y and z in L1,

<math>\left\{x, y\right\}[z]+\left\{y, z\right\}[x]+\left\{z, x\right\}[y]=0<math>

A * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z2 grading and satisfies [x,y]*=[y*,x*] for all x and y in the Lie superalgebra. Its universal enveloping algebra would be an ordinary *-algebra.

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