Super-algebras-Scholarly-Peer-review-Journal | Peer Reviewed Journals
Mathematica Eterna

Mathematica Eterna
Open Access

ISSN: 1314-3344


In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around). 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:{\displaystyle [x,y]=-(-1)^{|x||y|}[y,x].\ }[x,y]=-(-1)^{|x| |y|}[y,x].\ The super Jacobi identity: {\displaystyle (-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,}{\displaystyle (-1)^{|x||z|}[x,[y,z]]+(-1)^{|y||x|}[y,[z,x]]+(-1)^{|z||y|}[z,[x,y]]=0,} where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1). The degree of [x,y] is the sum of degree of x and y modulo 2.

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