Unitary Representation Of A Star 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

:$(c_1 A+c_2 B)\cdot X=c_1 A\cdot X + c_2 B\cdot X$

:$A\cdot (c_1 X + c_2 Y)=c_1 A\cdot X + c_2 A\cdot Y$

:$(-1)^=(-1)^A(-1)^X$

: ,Bcdot X=A\cdot (B\cdot X)-(-1)^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

: ,bsup>*= ^*,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 sup>*=-(-1)^LaL^{* *} 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 ψ>)(L , ψ>+(-1)^Laa(L embedding, embedded_within_A_in_the_sense_that_ther.html" ;"title="ψ>]).

Sometimes, the Lie superalgebra is embedding, 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)^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

Representation theory of Lie algebras
Supersymmetry

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Sometimes, the Lie superalgebra is embedding, 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)^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

Representation theory of Lie algebras
Supersymmetry

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