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An abelian Lie group has commutative multiplication. Its Lie algebra is an abelian Lie algebra, and its Lie exponential map is a group homomorphism from the additive Lie algebra. Conversely, an additive Lie exponential map forces the identity component of a Lie group to be abelian, but does not force all disconnected components to commute.

The Exponential map of a Lie group is a group homomorphism from the additive Lie algebra exactly when its identity component of a Lie group is abelian. If it is additive, its image is an abelian subgroup containing an identity neighborhood by the local exponential chart; connectedness makes that image the entire identity component. Conversely, in an abelian identity component the product $\exp (tX)\exp (tY)$ is a one-parameter subgroup with derivative $X+Y$, giving the identity by uniqueness. The disconnected Lie group $S_3\times \mathbb{R}$ shows that the full group need not be abelian.

When appropriate negative-square two-cycles of a K3 surface collapse, wrapped M2-branes become massless. Their intersection pattern is an ADE root diagram, and the massless charged states enlarge the abelian gauge algebra to the corresponding simply laced Lie algebra.

// Target: supersymmetry.bigb

The adjoint bundle is the associated vector bundle for the Adjoint representation of a Lie group on its Lie algebra. The gauge curvature of a principal connection is globally a two-form with values in this bundle. Differences of principal connections and their infinitesimal variations are one-forms with values in the same bundle.

Give the Lie algebra of a compact Lie group an inner product invariant under the Adjoint representation. For a maximal torus $T$ with Lie algebra $\mathfrak{t}$, its Lie-algebra centralizer is $\mathfrak{t}$: any additional commuting vector would generate, together with $T$, a larger compact connected abelian subgroup. The torus weight decomposition therefore supplies $H\in \mathfrak{t}$ whose centralizer is exactly $\mathfrak{t}$, by avoiding finitely many nonzero-weight hyperplanes. For any $X$, maximize $⟨{\mathrm{Ad}}_{g}X,H⟩$ on the group. Its directional derivative gives $⟨Z,[{\mathrm{Ad}}_{g}X,H]⟩=0$ for every $Z$, so ${\mathrm{Ad}}_{g}X\in \mathfrak{t}$. For a connected compact group, the bi-invariant Riemannian metric and Hopf-Rinow theorem make the exponential onto; exponentiating this conjugacy puts every element in a conjugate of $T$.

The adjoint representation is ${\mathrm{ad}}_x(y)=[x,y]$. For a semisimple Lie algebra, its nonzero weights are the roots and its zero-weight space is the Cartan subalgebra.

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.

The affine current algebra is the one-dimensional central extension of the loop algebra of a finite-dimensional Lie algebra, with central element acting as the level $k$. A restricted module has $J_{n}v=0$ for all sufficiently large $n$ for each vector $v$; this makes the Sugawara construction finite on each vector after normal ordering. In a compact unitary positive-energy representation, the nontrivial integrable levels are positive integers in the standard basic normalization.

The two-dimensional Lie algebra of infinitesimal dilations and translations of a line has a basis $h,e$ with $[h,e]=e$. Its derived algebra is the abelian line spanned by $e$, but every nontrivial term of its lower central series of a Lie algebra is that same line. It is solvable and not nilpotent.

An alternating trilinear form is a scalar-valued map, linear in three arguments, which vanishes whenever two arguments agree. In characteristic zero, interchanging two arguments negates its value, while cyclic permutation preserves it. For a symmetric bilinear form $B$ on a Lie algebra, the tensor $B([x,y],z)$ is alternating exactly when $B$ is an invariant bilinear form on a Lie algebra.

The relation $[x,y]=-[y,x]$ for a bilinear map proposed as a Lie bracket. An Alternating Lie bracket always satisfies it. Conversely, setting $y=x$ gives $2[x,x]=0$, so antisymmetry implies alternation over a field of characteristic different from two.

In characteristic two, the sign disappears and this relation just says $[x,y]=[y,x]$. It does not force $[x,x]=0$, even when the Jacobi identity holds. For example, on a two-dimensional vector space with basis $u,v$, set $[u,u]=v$ and every other basis bracket to zero, then extend bilinearly. Every nested bracket vanishes, so the Jacobi identity holds, and the product is symmetric. Nevertheless it is not a Lie algebra, because $[u,u]\ne 0$.

For $G_{\lambda }=\{g:\omega (g)\ge \lambda \}$ and $G_{\lambda +}=\{g:\omega (g)>\lambda \}$, the associated graded groupis a graded Lie algebra with bracket induced by the group commutator. For a p-valuation, $t{\mathrm{gr}}_{\lambda }(g)={\mathrm{gr}}_{\lambda +1}(g^p)$ makes it a graded ${\mathbb{F}}_p[t]$-Lie algebra.

For simple roots $(1,0)$ and $(-1,1)$, coroot integrality gives the weight lattice $P={\mathbb{Z}}^2\cup (\mathbb{Z}+1/2{)}^2$. This is the lattice for the Spin group $\mathrm{Spin}(5)$ and its Lie algebra. Representations descending to $SO(5)$ use the integer coset, because a $2\pi$ rotation in either coordinate plane must act trivially.

A nondegenerate invariant symmetric bilinear form on a Lie algebra extends by translation to a metric preserved by both left and right translation. It may have indefinite signature. Its Levi-Civita connection is ${\nabla }_{X}Y=\frac{1}{2}[X,Y]$ for left-invariant vector fields, and its Ricci tensor is $-K/4$. Choosing the nondegenerate Killing form of a semisimple Lie algebra yields a canonical indefinite Einstein manifold; compact groups instead admit the positive metric $-K$.

The average of the two flat translation connections on a Lie group is a torsion-free connection. For left-invariant vector fields, the Jacobi identity gives $R(X,Y)Z=-\frac{1}{4}[[X,Y],Z]$ with $R(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}$. Contracting gives $\mathrm{Ric}(X,Y)=-\frac{1}{4}K(X,Y)$ for the Killing form $K$. It is flat exactly when all double Lie brackets vanish, including the case of a two-step nilpotent group. For a semisimple Lie algebra, $K$ is nondegenerate and supplies a bi-invariant indefinite metric whose Levi-Civita connection is ${\nabla }^0$.

On a graded nilpotent Lie group whose first layer generates its Lie algebra, a horizontal curve obeys ${\dot{\gamma }}_t=(L_{{\gamma }_t}{)}_{\ast }{\dot{x}}_t$ for a control in the first layer. The displayed infimum is over horizontal curves with prescribed endpoints. The distance is left invariant and homogeneous under the graded dilations. On a free nilpotent Lie group it measures the least Euclidean length of a path realizing a prescribed truncated path signature.

A finite-dimensional complex Lie algebra is semisimple if and only if its Killing form is nondegenerate.

For a finite-dimensional Lie algebra over a field of characteristic zero, Cartan's criterion characterizes a solvable Lie algebra by vanishing of its Killing form on $\mathfrak{g}\times [\mathfrak{g},\mathfrak{g}]$, and a semisimple Lie algebra by nondegeneracy of that form. The Cartan solvability criterion and Cartan criterion for semisimplicity state the two distinct tests.