In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, a representation of a Lie superalgebra is an
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...
''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
:
:
:
:
Equivalently, a representation of ''L'' is a Z
2-graded representation of the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
of ''L'' which respects the third equation above.
Unitary representation of a star Lie superalgebra
A
* Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...
is a complex Lie superalgebra equipped with an
involutive antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
* such that * respects the grading and
:
,bsup>*=
*,a*
A
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
of such a Lie algebra is a Z
2 graded Hilbert space which is a representation of a Lie superalgebra as above together with the requirement that
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a st ...
elements of the Lie superalgebra are represented by
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
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 In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
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_there_is_a_homomorphism_from_the_
universal_enveloping_algebra_
In_mathematics,_the_universal_enveloping_algebra_of_a_Lie_algebra_is_the__unital__associative_algebra_whose__representations_correspond_precisely_to_the__representations_of_that_Lie_algebra.
Universal_enveloping_algebras_are_used_in_the__represent_...
aL.
This_approach_avoids_working_directly_with_a_Lie_supergroup,_and_hence_avoids_the_use_of_auxiliary_
of the Lie superalgebra to A. In that case, the equation above reduces to
:L
aL.
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary
of the Lie superalgebra to A. In that case, the equation above reduces to
:L
aL.
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary