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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 branching rule decomposes a representation into irreducible representations after restriction from a Lie group or Lie algebra to a subgroup or subalgebra.

A branching rule gives the irreducible representation decomposition after restricting a group representation to a subgroup, or a Lie algebra representation to a Lie subalgebra. For the standard block embedding $SU(5)\subset SU(6)$, $6\downarrow =5\oplus 1$ and ${\Lambda }^{2}6\downarrow =10\oplus 5$.

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 finite-dimensional complex Lie algebra representation and 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 nonzero 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.

Over a field of characteristic zero, in a finite-dimensional Lie algebra representation of an sl2 triple, start with a nonzero vector $X_0$ satisfying $f{X}_0=0$ and $h{X}_0=-\lambda X_0$, and set $X_n=e^n{X}_0$. The commutator identities give the displayed weight and lowering formulas by induction. If $X_N\ne 0$ and $X_{N+1}=0$, then $f{X}_{N+1}=(N+1)(\lambda -N)X_N$ forces $\lambda =N$, a nonnegative integer. The nonzero terminal vector is essential: choosing a later index after the ladder has already vanished gives no constraint. Applied to the Adjoint representation of a semisimple Lie algebra, this argument proves integrality of Cartan integers.

For a Lie algebra representation $V={⨁}_{\mu }{V}_{\mu }$ admitting a weight decomposition with finite-dimensional weight spaces, its formal character records the weight multiplicities:The symbols $e^{\mu }$ distinguish weights; they are not numerical exponential evaluations. A finite-dimensional integral Lie algebra representation has its formal character in the group ring of a weight lattice. For a Verma module $M(\lambda )$, the weights instead lie in $\lambda -Q_{+}$, where $Q_{+}$ is the nonnegative integer span of simple roots. An arbitrary highest weight $\lambda$ need not lie in the weight lattice, so this generally requires a formal completion of the corresponding weight coset, rather than the group ring of a weight lattice alone.

The Poincaré-Birkhoff-Witt theorem givesThe product is expanded as formal geometric series toward lower weights; each coefficient is finite and equals the Kostant partition function. Specifying this support condition makes the infinite expression meaningful without assuming analytic convergence.

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 Spin group $\mathrm{Spin}(7)$ double covers SO(7). Its central kernel element $-1$ acts trivially in the vector representation and as $-I$ in the spinor representation. Thus a spinor Lie algebra representation does not define an ordinary SO(7) group representation. Under $\mathrm{Spin}(6)\cong SU(4)$ the same obstruction is visible because $-I_4$ acts trivially on ${\Lambda }^{2}4$ but negatively on $4\oplus \overline{4}$.

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.