The multiplication $(x^{\prime },y^{\prime },z^{\prime })(x,y,z)=(x^{\prime }+e^{-z^{\prime }}x,y^{\prime }+e^{z^{\prime }}y,z^{\prime }+z)$ defines the three-dimensional Sol Lie group. Its left-invariant vector fields $e^{-z}{\partial }_x,e^z{\partial }_y,{\partial }_z$ have dual coframe $e^{z}dx,e^{-z}dy,dz$. Their sum of squares is a left-invariant metric, $d{s}^2=e^{2z}d{x}^2+e^{-2z}d{y}^2+d{z}^2$.

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.

A complex Lie group is a complex manifold whose multiplication and inversion are holomorphic maps between complex manifolds. Its tangent space at the identity is a complex Lie algebra; the Adjoint representation of a Lie group and the Lie exponential map are holomorphic. Compatibility with a complex manifold structure is essential: an arbitrary even-dimensional real Lie group need not have holomorphic multiplication.

The universal cover of a connected Lie group carries a unique lifted Lie group structure with a chosen identity, making its covering map a Lie group homomorphism. The kernel is a discrete central subgroup: conjugation of any kernel element is a continuous map from a connected group into a discrete set, hence constant. Every connected Lie group is therefore a quotient of its simply connected cover by a discrete central subgroup.

Every finite-dimensional real Lie algebra is the Lie algebra of a connected simply connected Lie group. One construction first uses the Ado theorem to embed it into a matrix Lie algebra, integrates the corresponding left-invariant distribution by the Frobenius theorem to a connected immersed Lie subgroup, and then takes a universal covering Lie group. The immersed subgroup need not be closed; taking its ambient closure may change its Lie algebra.

Differentiation sends Lie groups to Lie algebras and smooth group homomorphisms to bracket-preserving linear maps. On connected simply connected Lie groups, this is an equivalence with finite-dimensional real Lie algebras: the Lie third theorem gives existence, and integration of a Lie algebra homomorphism gives unique maps. General connected groups require discrete central quotient data, and disconnected groups additionally require component and conjugation data.

$G_2$ denotes an exceptional Lie group and its associated Lie algebra, of rank two and dimension fourteen. Its roots form the G2 root system. The compact real form is the automorphism group of the octonion algebra; other real forms and the complex Lie algebra have the same complex root-system type.

Choose an injective local exponential chart on a ball of radius $r$ in the Lie algebra, and let $0<\epsilon <r/2$. The neighborhood $\exp (B_{\epsilon })$ contains no nontrivial subgroup. Otherwise it contains an element $\exp X$ with $0<\Vert X\Vert <\epsilon$. Choose the first integer $m$ with $m\Vert X\Vert \ge \epsilon$. Then $\epsilon \le m\Vert X\Vert <2\epsilon <r$, so chart injectivity puts $(\exp X{)}^m=\exp (mX)$ outside $\exp (B_{\epsilon })$, a contradiction. Zero-dimensional Lie groups have the singleton identity as such an open neighborhood.

A Lie algebra over the p-adic integers whose underlying module is free of finite rank and whose Lie bracket satisfies $[L,L]\subseteq p^{ϵ}L$, where $ϵ=1$ for odd $p$ and $ϵ=2$ for $p=2$. Finite freeness is part of this convention: merely requiring a small Lie bracket on a torsion module would not correspond to uniform pro-p groups.

A real Lie algebra is compact when it is the Lie algebra of some compact Lie group. Equivalently, it has a positive-definite invariant bilinear form on a Lie algebra. It is a direct sum of an abelian center and a compact semisimple Lie algebra. For its semisimple part, the compactness criterion from the Killing form is negative-definiteness of the Killing form. This definition concerns the algebra; a global group with that algebra can still have a noncompact central covering direction.

For a Lie algebra $\mathfrak{g}$ over a field $k$ and a Lie algebra representation $V$, set $C^n(\mathfrak{g},V)={\mathrm{Hom}}_k({\Lambda }^n\mathfrak{g},V)$. These alternating multilinear maps form a cochain complex with differentialThe hat means that the argument is omitted. The Jacobi identity and the Lie algebra representation identity give $d^2=0$. The Lie algebra cohomology is the quotient vector space $H^n(\mathfrak{g},V)=\ker (d:C^n\to C^{n+1})/\mathrm{im}(d:C^{n-1}\to C^n)$. In degree zero it consists of the invariant vectors of a Lie algebra representation; with a trivial Lie algebra representation as coefficients, $H^1(\mathfrak{g},k)={\mathrm{Hom}}_k(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}],k)$.^[ref]

For trivial real coefficients, $H^2(\mathfrak{g},\mathbb{R})$ classifies central extensions of a Lie algebra by a fixed one-dimensional center, up to equivalences inducing the identity on the quotient and center. If $\mathfrak{g}$ is abelian and two-dimensional, every alternating bilinear form is a Lie algebra two-cocycle and every coboundary is zero; a nonzero such form produces the Heisenberg Lie algebra.

If $D:\mathfrak{b}\to \mathrm{Der}(\mathfrak{a})$ is a Lie algebra homomorphism into the derivations of a Lie algebra, the semidirect product has underlying vector space $\mathfrak{a}\oplus \mathfrak{b}$ and bracket $[(a,b),(a^{\prime },b^{\prime })]=([a,a^{\prime }]+D_b{a}^{\prime }-D_{b^{\prime }}a,[b,b^{\prime }])$. The homomorphism and derivation identities ensure the Jacobi identity. The first factor is an ideal; the second is a subalgebra whose action records the twisting. The diamond Lie algebra arises by letting a one-dimensional algebra act on a Heisenberg Lie algebra.

// Target: linear-algebra.bigb

The Lie algebra over a commutative ring $R$ generated by a set $X$ subject only to bilinearity, alternation and the Jacobi identity. Every map from $X$ to the underlying set of an $R$-Lie algebra extends uniquely to a Lie algebra homomorphism. It can be constructed from bracketed words by quotienting by those identities; elements are represented by Lie polynomials.

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra, with a chosen normalized invariant bilinear form on a Lie algebra $B$. Its affine current algebra is the central extension of a Lie algebra $\widehat{L\mathfrak{g}}=(\mathfrak{g}\otimes \mathbb{C}[t,t^{-1}])\oplus \mathbb{C}K$ of the loop algebra, withInvariance of $B$ makes the extra term a Lie algebra two-cocycle, hence preserves the Jacobi identity. In a representation where $K=kI$, the scalar $k$ is the level. The central term vanishes for zero modes, which recover the original Lie algebra, but modifies the commutator of opposite nonzero modes.^[ref]

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. Nontrivial unitary integrable positive-energy highest-weight representations have positive integer levels in the basic normalization. The zero-level integrable highest-weight representation is trivial. Adjoining the degree derivation gives the further extension with $[d,x\otimes t^n]=nx\otimes t^n$.

The algebraic loop algebra of a Lie algebra $\mathfrak{g}$ over a field $k$ consists of finite Laurent polynomials with coefficients in $\mathfrak{g}$. Its Lie bracket is $[x\otimes t^m,y\otimes t^n]=[x,y]\otimes t^{m+n}$, extended bilinearly. Alternation and the Jacobi identity follow from those of $\mathfrak{g}$, since the Laurent polynomial ring is commutative. Appropriate smooth or formal completions give other loop-algebra variants; the affine current algebra adds a central extension to these Fourier modes. For the affine extension, see Kac and Wakimoto, Section 0.2.

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.

Equip the Boolean hypercube with uniform probability measure and graph distance divided by its graph diameter. These spaces form a Lévy family of graphs. The vertex-isoperimetric inequality in the discrete cube reduces the complement of a positive-density set's radius-$\epsilon n$ graph neighbourhood to a binomial distribution tail. For fixed density $\delta >0$, Chebyshev's inequality puts a lower density quantile at least $n/2-C_{\delta }\sqrt{n}-O(1)$. After enlarging by $\epsilon n$, the omitted mass is at most $n/[4(\epsilon n-C_{\delta }\sqrt{n}-O(1){)}^2]$, which tends to zero.

Equip $[n{]}^d$ with uniform probability measure and its nearest-neighbour graph distance divided by $d(n-1)$, for fixed positive $d$. The set $\{x:x_1\le \lceil n/2\rceil \}$ has at least half the vertices. Its radius-$\epsilon d(n-1)$ graph neighbourhood still omits a proportion tending to $1/2-\epsilon d$ when $0<\epsilon <1/(2d)$. These grids therefore do not form a Lévy family of graphs.

For a probability measure on the real line with characteristic function $\phi$, averaging $1-\cos (tx)$ over $[-h,h]$ gives $1-\sin (hx)/(hx)$. This is at least $1/2$ when $|x|\ge 2/h$. Integrating gives the displayed bound. Consequently pointwise convergence of characteristic functions to a function continuous at zero implies tightness of probability measures, by the dominated convergence theorem on $[-h,h]$ and a separate tail bound for the finitely many initial measures.

On a trivialized fiber bundle over $U$, assume the fiber has finite-dimensional total cohomology over a field. The Künneth theorem identifies the total-space cohomology with a finite free graded module over $H^{\ast }(U)$. Classes restricting to a basis on every fiber give an invertible change of generators: its degree-zero coefficient blocks are invertible on each path component, while its positive-base-degree terms strictly decrease fiber degree and are therefore nilpotent. Inverting the diagonal blocks and then using a finite geometric series proves that multiplication by the chosen classes is an isomorphism over $U$.