Representation of a Lie superalgebra

In mathematics, particularly in the theory of Lie superalgebras, a representation of a Lie superalgebra L is the action of L upon a Z₂-graded vector space V such that if A and B are any two pure elements of L (remember that L is Z₂-graded) and X and Y are any two pure elements of V, then

$(c_1 A+c_2 B)[X]=c_1 A[X] + c_2 B[X]\,$

$A[c_1 X + c_2 Y]=c_1 A[X] + c_2 A[Y]\,$

$(-1)^{A[X]}=(-1)^A(-1)^X\,$

$[A,B)[X]=A[B[X]]-(-1)^{AB}B[A[X]].\,$

Equivalently, a representation of L is a Z₂-graded representation of the universal enveloping algebra of L which respects the third equation above.

Representation of a Lie superalgebra

See also

See also: Representation of a Lie superalgebra, Graded vector space, Group representation, Lie superalgebra, Mathematics, Pure element, Representation of a Hopf algebra, Representation of a Lie algebra, Universal enveloping algebra