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

${\displaystyle (c_{1}A+c_{2}B)\cdot X=c_{1}A\cdot X+c_{2}B\cdot X}$

${\displaystyle A\cdot (c_{1}X+c_{2}Y)=c_{1}A\cdot X+c_{2}A\cdot Y}$

${\displaystyle (-1)^{A\cdot X}=(-1)^{A}(-1)^{X}}$

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

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.

See also

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