NationMaster - Encyclopedia: Representation of a Lie algebra

In mathematics, a 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

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. In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint...

Formal definition

A representation of a Lie algebra $mathfrak g$ is a Lie algebra homomorphism In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

from $mathfrak g$ to the Lie algebra of endomorphisms on a vector space

V

(with the commutator as the Lie bracket). Explicitly, this means that In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

for all

x, y

in $mathfrak g$ . The vector space

V

, together with the representation

ρ

, is called a $mathfrak g$ -module. (Many authors abuse terminology and refer to

V

itself as the representation).

One can equivalently define a $mathfrak g$ -module as a vector space

V

together with a bilinear map $mathfrak g times Vto V$ such that In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

for all

x, y

in $mathfrak g$ and

v

V

. This is related to the previous definition by setting $xcdot v = rho_x(v).$

Infinitesimal Lie group representations

If $phi:Gto H$ is a homomorphism of Lie groups, and $mathfrak g$ and $mathfrak h$ are the Lie algebras of

G

and

H

respectively, then the induced map $phi_*: mathfrak g to mathfrak h$ on tangent spaces is a Lie algebra homomorphism. In particular, a representation of Lie groups In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... Suppose that Ï† : M â†’ N is a smooth map between smooth manifolds; then the differential of Ï† at a point x is, in some sense, the best linear approximation of Ï† near x. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ...

from $mathfrak g$ to the Lie algebra of the general linear group

GL(V)

, i.e. the endomorphism algebra of

V

. In mathematics, the general linear group of degree n is the set of nÃ—n invertible matrices, together with the operation of ordinary matrix multiplication. ...

A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

Properties

Representations of a Lie algebra are in one-to-one correspondence with representations of the associated universal enveloping algebra. This follows from the universal property of that construction. Image File history File links Broom_icon. ... In abstract algebra, a representation of an associative algebra is a module for that algebra. ... In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

If the Lie algebra is semisimple, then all reducible representations are decomposable. Otherwise, that's not true in general. In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... Reduction Formula We use the technique of integration by parts to evaluate a whole class of integrals by reducing them to simpler forms. ... In point-set topology, a continuum is decomposable if and only if it is the union of two of its proper subcontinua. ...

If we have two representations, with V₁ and V₂ as their underlying vector spaces and ·[·]₁ and ·[·]₂ as the representations, then the product of both representations would have $V_1otimes V_2$ as the underlying vector space and

If L is a real Lie algebra and $rho:Ltimes Vrightarrow V$ 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 $(zomega)[X]=bar{z}omega[X]$ 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]. In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...

See also

Categories: Cleanup from December 2005 | All pages needing cleanup | Representation theory of Lie algebras | Lie algebras Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups. ... In the theory of Lie superalgebras, a representation of a Lie superalgebra L is the action of L upon a Z2-graded vector space V such that if A and B are any two pure elements of L (remember that L is Z2-graded) and X and Y are any... In mathematics, there is a concept of a representation of a Hopf algebra. ... In abstract algebra, a representation of an associative algebra is a module for that algebra. ... If we have a Lie superalgebra L, then, a (not necessarily associative) Z2 gradedalgebra A is an algebra representation of L if as a Z2graded vector space, A is a vector space rep of L and in addition, the elements of L acts as derivations/antiderivations. ... If we have a Lie superalgebra L, then, a (not necessarily associative) Z2 gradedalgebra A is an algebra representation of L if as a Z2graded vector space, A is a vector space rep of L and in addition, the elements of L acts as derivations/antiderivations. ... You know what an ordinary (vector space) representation of a Hopf algebra is. ... 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 (Hausdorff) topological... In mathematics, a * Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map * such that * respects the grading and [a,b]*=[b*,a*] A unitary representation of such a Lie algebra is a Z2 graded Hilbert space which is a representation of the Lie superalgebra together with...

Contents

Formal definition GA_googleFillSlot("encyclopedia_square");

Infinitesimal Lie group representations

Properties

See also

Formal definition