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Representation of a Lie superalgebra

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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 {\displaystyle (c_{1}A+c_{2}B)\cdot X=c_{1}A\cdot X+c_{2}B\cdot X}
A ⋅ ( c 1 X + c 2 Y ) = c 1 A ⋅ X + c 2 A ⋅ Y {\displaystyle A\cdot (c_{1}X+c_{2}Y)=c_{1}A\cdot X+c_{2}A\cdot Y}
( − 1 ) A ⋅ X = ( − 1 ) A ( − 1 ) X {\displaystyle (-1)^{A\cdot X}=(-1)^{A}(-1)^{X}}
[ A , B ] ⋅ X = A ⋅ ( B ⋅ X ) − ( − 1 ) A B B ⋅ ( A ⋅ X ) . {\displaystyle \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 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*=-(-1)L*) 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=(L)|ψ>+(-1)a(L).

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=La-(-1)aL.

This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary Grassmann numbers.

See also


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