Representation of a Lie superalgebra
Encyclopedia
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 algebra
of L which respects the third equation above.
is a complex Lie superalgebra equipped with an involutive antilinear map
* such that * respects the grading and
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[a]*=-(-1)LaL*[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,
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
This approach avoids working directly with a Lie supergroup, and hence avoids the use of auxiliary Grassmann number
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of representation theory
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 studiesmodules over these abstract algebraic structures...
, a representation of a Lie superalgebra is an action of Lie superalgebra
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...
L on a Z2-graded vector space
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
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 algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of L which respects the third equation above.
Unitary representation of a star Lie superalgebra
A * Lie superalgebraLie 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...
is a complex Lie superalgebra equipped with an involutive antilinear map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....
* such that * respects the grading and
- [a,b]*=[b*,a*].
A unitary representation
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 is a unitary operator for every g ∈ G...
of such a Lie algebra is a Z2 graded
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
which is a representation of a Lie superalgebra as above together with the requirement that self-adjoint
Self-adjoint
In mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
elements of the Lie superalgebra are represented by Hermitian
Hermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...
transformations.
This is a major concept in the study of supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
together with representation of a Lie superalgebra on an algebra. Say A is an *-algebra
Star-algebra
-*-ring:In mathematics, a *-ring is an associative ring with a map * : A → A which is an antiautomorphism and an involution.More precisely, * is required to satisfy the following properties:* ^* = x^* + y^** ^* = y^* x^** 1^* = 1...
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 representation
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 is a unitary operator for every g ∈ 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[a|ψ>)]=(L[a])|ψ>+(-1)Laa(L[|ψ>]).
Sometimes, the Lie superalgebra is embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
within A in the sense that there is a homomorphism from the universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of 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 Grassmann number
Grassmann number
In mathematical physics, a Grassmann number, named after Hermann Grassmann, is a mathematical construction which allows a path integral representation for Fermionic fields...
s.
See also
- Graded vector spaceGraded vector spaceIn mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
- Lie algebra representation
- Representation theory of Hopf algebras