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Every finite-dimensional Lie algebra over a field of characteristic zero has a finite-dimensional faithful Lie algebra representation. Thus it embeds as a Lie subalgebra of a matrix algebra, using the matrix commutator as bracket. Faithfulness is crucial for detecting central elements: the Adjoint representation alone kills the center of a Lie algebra.

A complete flag in an $n$-dimensional vector space is a nested sequence of vector subspaces with $\dim V_i=i$. An action preserves such a flag exactly when its matrices are upper triangular in a basis adapted to the flag. The Lie theorem supplies an invariant complete flag for every complex finite-dimensional Lie algebra representation of a solvable Lie algebra.

A composition factor is an irreducible successive quotient in a composition series of a module. For a central Casimir operator, its scalar on a composition factor is an eigenvalue of the operator on the original Lie algebra representation.

A composition series of a module is a finite chain of submodules whose nonzero successive quotient modules are simple modules. Every finite-dimensional Lie algebra representation has such a chain, viewed as a module over the universal enveloping algebra.

The derived representation of a smooth Lie-group representation is $dD(X)={\frac{d}{dt}|}_{0}D(\exp (tX))$. It is a Lie algebra representation because $dD([X,Y])=[dD(X),dD(Y)]$.

For a smooth homomorphism of Lie groups, naturality gives $f(\exp X)=\exp (d{f}_{e}X)$. Therefore $\log (f(g))=d{f}_e(\log g)$ locally. Apply this to the group commutator and differentiate its two curve parameters to obtain the displayed identity for the Lie bracket from local group commutators. In particular the differential of a group representation is a Lie algebra representation.

The dual of a Lie algebra representation on $V$ acts on its dual space by $(x\varphi )(v)=-\varphi (xv)$. The minus sign makes this a Lie algebra homomorphism: taking transposes reverses the order in a commutator. It is the infinitesimal version of the dual representation of a group.

A finite-dimensional Lie algebra representation in which every representing endomorphism is nilpotent has a nonzero vector annihilated by the entire Lie algebra. Applying this to the Adjoint representation is the key step in Engel theorem.

A Lie algebra representation on $V$ induces one on each exterior power by acting on every factor and summing. The tensor product of Lie algebra representations preserves the defining alternating relations, so this descends to the quotient in every characteristic of a field. In characteristic two define the exterior algebra by the relations $v\wedge v=0$, rather than by dividing a tensor antisymmetrizer by $2$.

A Lie algebra representation is faithful when its representing homomorphism is injective.

With $D_{\mu }={\partial }_{\mu }+R(A_{\mu })$ and ${\delta }_X{A}_{\mu }=-ϵ{\partial }_{\mu }X+ϵ[X,A_{\mu }]$, the derivative terms in ${\delta }_X(D_{\mu }\varphi )$ cancel. The remaining terms combine using $R([X,A_{\mu }])=[R(X),R(A_{\mu })]$, so the covariant derivative transforms in the same Lie algebra representation as the scalar. This convention absorbs the gauge coupling into the connection.

Every finite-dimensional complex Lie algebra representation of a nilpotent Lie algebra decomposes into invariant generalized weight spaces. The nilpotence assumption is essential: a general solvable action can have extensions between distinct weights.

Here is a proof. For $h,k\in H$, nilpotence gives $(\mathrm{ad}h{)}^{N}k=0$, hence $(\mathrm{ad}\rho (h){)}^N\rho (k)=0$. By the nilpotent commutator preserves generalized eigenspaces lemma, every generalized eigenspace of $\rho (h)$ is invariant under every $\rho (k)$. Split successively by the acting operators of a basis of $H$. Each resulting block is invariant under $H$, and each basis operator has only one eigenvalue on it. The Lie theorem triangularizes the action on each block; its diagonal entries therefore give one and the same linear functional $\lambda$ on every basis vector of that block. Every $\rho (h)-\lambda (h)I$ is strictly upper triangular there, proving that the block is $V^{\lambda }$. Distinct blocks have distinct weights, so this is a direct sum. On the zero generalized weight space, every acting operator is a nilpotent endomorphism, so the Engel theorem supplies a nonzero common annihilated vector whenever that space is nonzero.

For a finite-dimensional complex Lie algebra representation $\rho$ of a nilpotent Lie algebra $H$, a generalized weight space is the simultaneous generalized eigenspace displayed above, where $\lambda :H\to \mathbb{C}$ is a linear functional. Every occurring $\lambda$ vanishes on the derived algebra of $H$. Ordinary weight vectors satisfy the same equations without the powers.

The group ring of a weight lattice $P$ consists of finite sums ${\sum }_{\lambda }{a}_{\lambda }e(\lambda )$ with integer coefficients, where $e(\lambda )e(\mu )=e(\lambda +\mu )$. A finite-dimensional Lie algebra representation has formal character ${\sum }_{\lambda }(\dim V_{\lambda })e(\lambda )$ in this ring. The Weyl group acts by $e(\lambda )\mapsto e(w\lambda )$. The ambient ring need not consist of genuine characters; the representation ring of a semisimple Lie algebra of finite-dimensional modules for a complex semisimple Lie algebra identifies with the invariant subring $\mathbb{Z}[P{]}^W$.

// Target: semisimple-lie-algebra.bigb

For Lie algebra representations $V$ and $W$, the Hom representation on $\mathrm{Hom}(V,W)$ isIts invariant vectors are exactly the intertwining maps.

For a smooth left Lie group action, define the fundamental vector field by $X^{\#}(p)={\frac{d}{ds}|}_0\exp (sX)\cdot p$. With the Lie bracket of vector fields $[U,V]f=U(Vf)-V(Uf)$, this map is an anti-homomorphism. The map $X\mapsto v_X=-X^{\#}$ is a Lie algebra representation. A right action with the positive exponential has the opposite homomorphism convention, so specifying action side and bracket convention prevents a sign ambiguity.

For a finite-dimensional real Lie algebra, a finite-dimensional Lie algebra representation integrates uniquely to a smooth representation of the connected simply connected Lie group with that Lie algebra. For another connected group with the same Lie algebra, integration is possible exactly when the representation of its universal cover is trivial on the covering kernel. This separates local Lie-algebra classification from global representation restrictions.

If a submodule $U$ of a Lie algebra representation $V$ admits an intertwining map $P:V\to U$ with $P{|}_U=I_U$, then $P^2=P$ and its kernel is an invariant complement. For a complex semisimple Lie algebra, the Hom representation on maps $V\to U$ whose restriction is a scalar identity reduces finding $P$ to splitting of a trivial quotient for a semisimple Lie algebra.

A nonzero Lie algebra representation is irreducible when it has no proper nonzero invariant subspace.

The Jacobi identity is the compatibility condition on a Lie bracketTogether with an Alternating Lie bracket, it defines a Lie algebra. Unlike associativity, it relates three cyclic bracketings rather than equating two bracketings. Equivalently, using antisymmetry of a Lie bracket,Thus the linear map ${\mathrm{ad}}_x:y\mapsto [x,y]$ is a derivation of a Lie algebra: acting on a bracket is the sum of acting on each argument. This makes infinitesimal symmetries act consistently on each other's products. It also gives $[{\mathrm{ad}}_x,{\mathrm{ad}}_y]={\mathrm{ad}}_{[x,y]}$, the identity that makes the Adjoint representation of a Lie algebra a Lie algebra representation.

For the commutator in an associative algebra, the cancellation is directly visible:Each word occurs once with each sign. This proves the Jacobi identity without a restriction on the characteristic of a field.

For a computed check in the sl2 matrix commutator example,Here the last two terms are individually nonzero over characteristic zero; the identity records their cancellation. In the cross-product Lie algebra, the same cyclic cancellation is the vector triple product identity.

The historical connection is unusually concrete: Lie and Engel's volume I expands nested commutators of differential operators into twelve terms and cancels them cyclically, in Chapter 5, §26, equation (7). See Lie and Engel, §26, in Merker's translation, pp. 105–106.