Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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In the 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. In the language of physics, one looks for a vector space ${\displaystyle V}$ together with a collection of operators on ${\displaystyle V}$ satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.

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 the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The universality of this ring says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra.

Formal definition

Let ${\displaystyle {\mathfrak {g}}}$ be a Lie algebra and let ${\displaystyle V}$ be a vector space. We let ${\displaystyle {\mathfrak {gl}}(V)}$ denote the space of endomorphisms of ${\displaystyle V}$, that is, the space of all linear maps of ${\displaystyle V}$ to itself. We make ${\displaystyle {\mathfrak {gl}}(V)}$ into a Lie algebra with bracket given by the commutator: ${\displaystyle =\rho \circ \sigma -\sigma \circ \rho }$ for all ρ,σ in ${\displaystyle {\mathfrak {gl}}(V)}$. Then a representation of ${\displaystyle {\mathfrak {g}}}$ on ${\displaystyle V}$ is a Lie algebra homomorphism

${\displaystyle \rho \colon {\mathfrak {g}}\to {\mathfrak {gl}}(V)}$.

Explicitly, this means that ${\displaystyle \rho }$ should be a linear map and it should satisfy

${\displaystyle \rho ()=\rho (X)\rho (Y)-\rho (Y)\rho (X)}$

for all X, Y in ${\displaystyle {\mathfrak {g}}}$. The vector space V, together with the representation ρ, is called a ${\displaystyle {\mathfrak {g}}}$-module. (Many authors abuse terminology and refer to V itself as the representation).

The representation ${\displaystyle \rho }$ is said to be faithful if it is injective.

One can equivalently define a ${\displaystyle {\mathfrak {g}}}$-module as a vector space V together with a bilinear map ${\displaystyle {\mathfrak {g}}\times V\to V}$ such that

${\displaystyle \cdot v=X\cdot (Y\cdot v)-Y\cdot (X\cdot v)}$

for all X,Y in ${\displaystyle {\mathfrak {g}}}$ and v in V. This is related to the previous definition by setting X ⋅ v = ρ(X)(v).

Examples

Adjoint representations

The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ on itself:

${\displaystyle {\textrm {ad}}:{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}}),\quad X\mapsto \operatorname {ad} _{X},\quad \operatorname {ad} _{X}(Y)=.}$

Indeed, by virtue of the Jacobi identity, ${\displaystyle \operatorname {ad} }$ is a Lie algebra homomorphism.

Infinitesimal Lie group representations

A Lie algebra representation also arises in nature. If ${\displaystyle \phi }$: G → H is a homomorphism of (real or complex) Lie groups, and ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle {\mathfrak {h}}}$ are the Lie algebras of G and H respectively, then the differential ${\displaystyle d_{e}\phi :{\mathfrak {g}}\to {\mathfrak {h}}}$ on tangent spaces at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space V, a representation of Lie groups

${\displaystyle \phi :G\to \operatorname {GL} (V)\,}$

determines a Lie algebra homomorphism

${\displaystyle d\mathrm{\varphi <!--\; \varphi \; -->}:\mathfrak{g}\to <!--\; \to \; -->\mathfrak{g}\mathfrak{l}(V)}$Zdroj:https://en.wikipedia.org?pojem=Representations_of_Lie_algebras
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