- Representation of a 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
Equivalently, a representation of "L" is a Z2-graded representation of the
universal enveloping algebraof "L" which respects the third equation above.
Unitary representation of a star Lie superalgebra
Lie superalgebrais a complex Lie superalgebra equipped with an involutive antilinear map* such that * respects the grading and
: [a,b] *= [b*,a*]
unitary representationof such a Lie algebra is a Z2 graded Hilbert spacewhich is a representation of a Lie superalgebra as above together with the requirement that self-adjointelements of the Lie superalgebra are represented by Hermitiantransformations.
This is a major concept in the study of
supersymmetrytogether with representation of a Lie superalgebraon an algebra. Say A is an *-algebra representation of the Lie superalgebra (together with the additional requirement that * respects the grading and L [a] *=-(-1)LaL* [a*] ) and H is the unitary rep and also, H is a unitary representationof 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)Laa(L [|ψ>] )
Sometimes, the Lie superalgebra is embedded within A in the sense that there is a homomorphism from the
universal enveloping algebraof the Lie superalgebra to A. In that case, the equation above reduces to
:L [a] =La-(-1)LaaL
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary
Graded vector space
Lie algebra representation
Representation theory of Hopf algebras
Wikimedia Foundation. 2010.
Look at other dictionaries:
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,… … Wikipedia
Lie algebra representation — Lie groups … Wikipedia
Representation theory — This article is about the theory of representations of algebraic structures by linear transformations and matrices. For the more general notion of representations throughout mathematics, see representation (mathematics). Representation theory is… … Wikipedia
Lie algebra — In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term… … Wikipedia
List of representation theory topics — This is a list of representation theory topics, by Wikipedia page. See also list of harmonic analysis topics, which is more directed towards the mathematical analysis aspects of representation theory. Contents 1 General representation theory 2… … Wikipedia
Graded Lie algebra — In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket… … Wikipedia
Restricted representation — In mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand … Wikipedia
List of Lie groups topics — This is a list of Lie group topics, by Wikipedia page. Contents 1 Examples 2 Lie algebras 3 Foundational results 4 Semisimple theory … Wikipedia
List of Lie group topics — This is a list of Lie group topics, by Wikipedia page. Examples See Table of Lie groups for a list *General linear group, special linear group **SL2(R) **SL2(C) *Unitary group, special unitary group **SU(2) **SU(3) *Orthogonal group, special… … Wikipedia
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 is a locally compact… … Wikipedia