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

:$(c\_1\; A+c\_2\; B)\; [X]\; =c\_1\; A\; [X]\; +\; c\_2\; B\; [X]\; ,$

:$A\; [c\_1\; X\; +\; c\_2\; Y]\; =c\_1\; A\; [X]\; +\; c\_2\; A\; [Y]\; ,$

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

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

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

Unitary representation of a star Lie superalgebra

A ^* Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map ^* such that * respects the grading and

: [a,b] ^*= [b^*,a^*]

A unitary representation of such a Lie algebra is a Z₂ graded Hilbert space which is a representation of a Lie superalgebra as above together with the requirement that self-adjoint elements of the Lie superalgebra are represented by Hermitian transformations.

This is a major concept in the study of supersymmetry together with representation of a Lie superalgebra on an algebra. Say A is an *-algebra representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L [a] ^*=-(-1)^LaL^* [a^*] ) and H is the unitary rep and also, H is a unitary representation of A.

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.

ee also

* Graded vector space
* Lie algebra representation
* Representation theory of Hopf algebras