 Lie algebra representation

Lie groups General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)Exponential map
Adjoint representation of a Lie group
Adjoint representation of a Lie algebra
Killing form
Lie point symmetryStructure of semisimple Lie groupsDynkin diagrams
Cartan subalgebra
Root system
Real form
Complexification
Split Lie algebra
Compact Lie algebraRepresentation of a Lie group
Representation of a Lie algebrav · mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.
Contents
Formal definition
A representation of a Lie algebra is a Lie algebra homomorphism
from to the Lie algebra of endomorphisms on a vector space V (with the commutator as the Lie bracket), sending an element x of to an element ρ_{x} of .
Explicitly, this means that
for all x,y in . The vector space V, together with the representation ρ, is called a module. (Many authors abuse terminology and refer to V itself as the representation).
One can equivalently define a module as a vector space V together with a bilinear map such that
for all x,y in and v in V. This is related to the previous definition by setting
Infinitesimal Lie group representations
If is a homomorphism of Lie groups, and and are the Lie algebras of G and H respectively, then the induced map on tangent spaces is a Lie algebra homomorphism. In particular, a representation of Lie groups
determines a Lie algebra homomorphism
from to the Lie algebra of the general linear group GL(V), i.e. the endomorphism algebra of V.
A partial converse to this statement says that every representation of a finitedimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simplyconnected Lie groups are in onetoone correspondence with representations of their Lie algebras.
Properties
Representations of a Lie algebra are in onetoone correspondence with algebra representations of the associated universal enveloping algebra. This follows from the universal property of that construction.
If the Lie algebra is semisimple, then all reducible representations are decomposable. Otherwise, that's not true in general.
If we have two representations, with V_{1} and V_{2} as their underlying vector spaces and ·[·]_{1} and ·[·]_{2} as the representations, then the product of both representations would have as the underlying vector space and
If L is a real Lie algebra and is a complex representation of it, we can construct another representation of L called its dual representation as follows.
Let V^{∗} be the dual vector space of V. In other words, V^{∗} is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that for any z in C, ω in V^{∗} and X in V. This is usually rewritten as a contraction with a sesquilinear form 〈·,·〉. i.e. 〈ω,X〉 is defined to be ω[X].
We define as follows:
 〈(A)[ω],X〉 + 〈ω,ρA[X]〉 = 0,
for any A in L, ω in V^{∗} and X in V. This defines uniquely.
Classification
Finitedimensional representations of semisimple Lie algebras
For more details on this topic, see Weight (representation theory).Similarly to how semisimple Lie algebras can be classified, the finitedimensional representations of semisimple Lie algebras can be classified. This is a classical theory, widely regarded as beautiful, and a standard reference is (Fulton & Harris 1992).
Briefly, finitedimensional representations of a semisimple Lie algebra are completely reducible, so it suffices to classify irreducible (simple) representations. Semisimple Lie algebras are classified in terms of the weights of the adjoint representation, the socalled root system; in a similar manner all finitedimensional irreducible representations can be understood in terms of weights; see weight (representation theory) for details.
Representation on an algebra
If we have a Lie superalgebra L, then a representation of L on an algebra is a (not necessarily associative) Z_{2} graded algebra A which is a representation of L as a Z_{2} graded vector space and in addition, the elements of L acts as derivations/antiderivations.
More specifically, if H is a pure element of L and x and y are pure elements of A,
 H[xy] = (H[x])y + (−1)^{xH}x(H[y])
Also, if A is unital, then
 H[1] = 0
Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors.
A Lie (super)algebra is an algebra and it has an adjoint representation of itself. This is a representation on an algebra: the (anti)derivation property is the superJacobi identity.
If a vector space is both an associative algebra and a Lie algebra and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra.
See also
References
 Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: SpringerVerlag, ISBN 9780387974958, MR1153249, ISBN 9780387975276
 J.Humphreys, Introduction to Lie algebras and representation theory, Birkhäuser, 2000
Categories: Representation theory of Lie algebras
Wikimedia Foundation. 2010.
Look at other dictionaries:
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
Algebra representation — In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint… … Wikipedia
Compact Lie algebra — Lie groups … Wikipedia
Lie algebra cohomology — In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by Chevalley and Eilenberg (1948) in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie … Wikipedia
Lie group — Lie groups … Wikipedia
Semisimple Lie algebra — In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non abelian Lie algebras mathfrak g whose only ideals are {0} and mathfrak g itself. It is called reductive if it is the sum of a semisimple and an… … Wikipedia
Affine Lie algebra — In mathematics, an affine Lie algebra is an infinite dimensional Lie algebra that is constructed in a canonical fashion out of a finite dimensional simple Lie algebra. It is a Kac–Moody algebra whose generalized Cartan matrix is positive semi… … Wikipedia
Solvable Lie algebra — In mathematics, a Lie algebra g is solvable if its derived series terminates in the zero subalgebra. That is, writing for the derived Lie algebra of g, generated by the set of values [x,y] for x and y in g, the derived series … 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
Nilpotent Lie algebra — In mathematics, a Lie algebra is nilpotent if the lower central series becomes zero eventually. Equivalently, is nilpotent if … Wikipedia
18+© Academic, 20002024 Contact us: Technical Support, Advertising
Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx.Share the article and excerpts
Lie algebra representation
 Lie algebra representation

Lie groups General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)Exponential map
Adjoint representation of a Lie group
Adjoint representation of a Lie algebra
Killing form
Lie point symmetryStructure of semisimple Lie groupsDynkin diagrams
Cartan subalgebra
Root system
Real form
Complexification
Split Lie algebra
Compact Lie algebraRepresentation of a Lie group
Representation of a Lie algebra