In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, a representation of a Lie superalgebra is an

action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...

Lie superalgebra In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a \Z/2\Z grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. The notion of \Z/2\Z gra ...

''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

cdot 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 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 representa ...

of ''L'' which respects the third equation above.

Unitary representation of a star Lie superalgebra

A *

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 of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...

* such that * respects the grading and : ,b= *,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 the ca ...

of such a Lie algebra is a Z₂ graded

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

which is a representation of a Lie superalgebra as above together with the requirement that

self-adjoint In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements ...

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 me ...

transformations. This is a major concept in the study of

supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...

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)^LaL* * and H is the unitary rep and also, H is a

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

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 number In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra of a complex vector space. The special case of a 1-dimensional algebra is known a ...

Unitary representation of a star Lie superalgebra

See also

See also