Representation of a Lie superalgebra
In the mathematical field of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z2-graded vector space V, such that if A and B are any two pure elements of L and X and Y are any two pure elements of V, then
Equivalently, a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation above.
Unitary representation of a star Lie superalgebra
A *
- [a,b]*=[b*,a*].
A
This is a major concept in the study of
These three reps are all compatible if for pure elements a in A, |ψ> in H and L in the Lie superalgebra,
- L[a|ψ>)]=(L[a])|ψ>+(-1)Laa(L[|ψ>]).
Sometimes, the Lie superalgebra is embedded within A in the sense that there is a homomorphism from the universal enveloping algebra of the Lie superalgebra to A. In that case, the equation above reduces to
- L[a]=La-(-1)LaaL.
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary Grassmann numbers.
See also