Unitary Representations Of A Star Lie Superalgebra
In
mathematics
, 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
Z
2
graded
Hilbert space
which is a
representation of the Lie superalgebra
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
algebra representation of a Lie superalgebra
. Say A is an
*-algebra
representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L
a
*
=-(-1)
La
L
*
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)
La
a(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)
La
aL
The attractive feature of this approach is that there is no need to introduce (mysterious)
Grassmann numbers
.
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