Lie algebra ($\mathfrak{g}$)

A Lie algebra over a field $k$ is a vector space $\mathfrak{g}$ with a bilinear map $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$, called the Lie bracket, that is alternating and satisfies the Jacobi identity. Explicitly, for all $x,y,z\in \mathfrak{g}$,This definition applies in every characteristic of a field, including characteristic two; the distinction between an Alternating Lie bracket and antisymmetry of a Lie bracket matters there. See Pavel Etingof, Lie Groups and Lie Algebras, Definition 8.4 and Remark 8.5.

The motivating question is how to describe very small symmetries. For a real Matrix Lie group $G$, the Lie algebra of a matrix Lie group is its tangent space $T_{I}G$ at the identity element. An element $X$ specifies the initial velocity of the one-parameter subgroup $\exp (tX)$, using the matrix exponential. The Lie bracket records the first nonzero effect of changing the order of two small transformations. Even when first-order motions cancel, a second-order motion can remain: the small rotation commutator below leaves a rotation about a third axis.

This turns local symmetry questions into computations with linear maps. A basis and its structure constants of a Lie algebra determine every Lie bracket by bilinearity. For example, three matrices suffice for the sl2 matrix commutator example, while the familiar cross product describes the SO(3) Lie algebra. These descriptions help calculate rotations in three dimensions and build Lie algebra representations; the same bracket structure underlies angular momentum commutation relations. The striking feature is that a nonlinear family of motions has a local description by a vector space and one product. Global information still needs care: the Lie algebra alone does not distinguish the circle group from the additive real line.

Historically, the connection with differential equations is visible in Sophus Lie's Theorie der Transformationsgruppen, volume I (1888), prepared with Friedrich Engel. Chapter 6 interprets first-order differential operators as infinitesimal transformations, linking their action on functions to the transformations those functions admit. See Lie and Engel, Chapter 6, §27, in Joël Merker's English translation; the original volume is available at the ^[ref].

Call a Lie algebra pre-supersoluble if it is abelian or its quotient by its centre has a one-dimensional ideal. Every nonzero such algebra has nonzero abelianization over every field. To prove this, choose $x$ outside the centre with $Z(L)+Fx$ an ideal and write $[a,x]=\lambda (a)x+z(a)$ with $z(a)$ central. The Jacobi identity gives $\lambda ([a,b])=0$. If $\lambda \ne 0$, the derived algebra lies in its proper kernel. If $\lambda =0$, Jacobi gives $[[a,b],x]=0$, so the derived algebra lies in the proper centralizer of an element of a Lie algebra of $x$. The abelian case is immediate. This condition does not imply solvability: ${\mathfrak{sl}}_2(\mathbb{C})$ direct sum the affine Lie algebra of the line has zero centre and a one-dimensional ideal in the affine factor, but is not solvable.

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.

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.

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 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.

A finite linear combination of iterated Lie brackets in formal variables. It evaluates naturally in every Lie algebra, and every Lie algebra homomorphism preserves its evaluation. The homogeneous terms of the Baker--Campbell--Hausdorff formula are rational Lie polynomials; this permits the group law to be reconstructed from the bracket wherever the series converges.

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.

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The Chevalley-Eilenberg complex consists of alternating maps from ${\mathfrak{g}}^k$ to a Lie algebra representation $V$, with differential built from the bracket and module action. For trivial real coefficients, a two-cocycle is an alternating bilinear form $\kappa$ satisfying $\kappa ([X,Y],Z)+\kappa ([Y,Z],X)+\kappa ([Z,X],Y)=0$. Coboundaries have the form $b([X,Y])$, up to the differential's sign convention. Their quotient is $H^2(\mathfrak{g},\mathbb{R})$ and classifies central extensions of a Lie algebra.

For trivial coefficients in a field $k$, a Lie algebra two-coboundary is a bilinear map $\beta (x,y)=t([x,y])$ for a linear functional $t$ in the dual space ${\mathfrak{g}}^{\ast }$. Some conventions put a minus sign in this formula; the resulting vector subspace $B^2(\mathfrak{g},k)$ is the same. Each such map is a Lie algebra two-cocycle: it is an alternating bilinear form, and its cyclic expression is $t([[x,y],z]+[[y,z],x]+[[z,x],y])=0$ by the Jacobi identity. Replacing a linear section of a central extension of a Lie algebra by $s^{\prime }(x)=s(x)+t(x)z$ changes its Lie algebra two-cocycle by $-\beta$. Thus $H^2(\mathfrak{g},k)=Z^2(\mathfrak{g},k)/B^2(\mathfrak{g},k)$ in Lie algebra cohomology records exactly the obstruction to a section that is a Lie algebra homomorphism.

For trivial real coefficients, a Lie algebra two-cocycle is an alternating bilinear form satisfying $\kappa ([X,Y],Z)+\kappa ([Y,Z],X)+\kappa ([Z,X],Y)=0$. Adding a form $b([X,Y])$ does not change its Lie algebra cohomology class. The bracket $[(X,a),(Y,b)]=([X,Y],\kappa (X,Y))$ defines a central extension of a Lie algebra by $\mathbb{R}$ precisely when this cocycle identity holds. The constant discrepancy in a moment-map equivariance obstruction is such a cocycle.

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.

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.

On the set underlying a uniform pro-p group, define scalar multiplication by $\lambda x=x^{\lambda }$, and define addition and the Lie bracket byRoots are the unique roots in the appropriate power layers. The limits give a powerful Lie lattice. In a suitable complete normed group algebra, the p-adic logarithm identifies these operations with ordinary addition and the associative-algebra commutator.

The Baker--Campbell--Hausdorff formula converges integrally on a powerful Lie lattice. A degree-$m$ Lie term has bracket divisibility $p^{ϵ(m-1)}$; the standard integral coefficient bound loses at most $\lfloor (m-1)/(p-1)\rfloor$ powers of $p$. The remaining valuation tends to infinity. Associativity follows from the formal exponential identity, the identity is zero, the inverse is $-x$, and $x^{\ast n}=nx$. The resulting pro-p group is a uniform pro-p group, with power subgroups $p^{n}L$.

The vector space ${\mathbb{R}}^3$ with the cross product as Lie bracket, $[x,y]=x\times y$. Bilinearity and $x\times x=0$ give an Alternating Lie bracket; the vector triple product identity proves the Jacobi identity because the three expressions $x\times (y\times z)=y(x\cdot z)-z(x\cdot y)$ cancel cyclically, using the symmetric dot product.

Its relation to the SO(3) Lie algebra is explicit. For a vector $x=(x_1,x_2,x_3)$, define the skew-symmetric matrixThen $[\hat{x},\hat{y}]=\widehat{x\times y}$: acting on $v$, the left side is $x\times (y\times v)-y\times (x\times v)=(x\times y)\times v$. Every real $3\times 3$ skew-symmetric matrix occurs uniquely, so this hat map is a Lie algebra isomorphism. With the standard basis vectors $e_1,e_2,e_3$,and their matrix commutators are $[X,Y]=Z$, $[Y,Z]=X$, $[Z,X]=Y$. With active rotation matrices acting on column vectors, $\exp (t\hat{x})$ rotates through angle $t\Vert x\Vert$ about the oriented axis $x$ when $x\ne 0$. In particular $\exp (tZ)e_1=(\cos t,\sin t,0)$, fixing the positive-angle convention used in the small rotation commutator.

The map $x\mapsto -(i/2)x\cdot \sigma$ is also a Lie algebra isomorphism to the compact real SU(2) Lie algebra, using the Pauli matrix commutator identity. These two identifications concern Lie algebras; they do not identify the corresponding Lie groups globally.

The real Lie algebra $\mathfrak{iso}(2)$ is a semidirect product of planar translations with infinitesimal rotations. One orientation convention is $[J,E_1]=-E_2$, $[J,E_2]=E_1$, $[E_1,E_2]=0$. Its translation space is a two-dimensional abelian ideal of a Lie algebra.

In the ordered basis $(J,E_1,E_2)$ of the Euclidean motion Lie algebra in two dimensions, the Killing form has matrix $\mathrm{diag}(-2,0,0)$. Its radical is the translation ideal, and the algebra is solvable rather than semisimple.

A graded Lie algebra has a graded Lie bracket preserving total degree, with $[u,v]=-(-1{)}^{|u||v|}[v,u]$ and $[u,[v,w]]=[[u,v],w]+(-1{)}^{|u||v|}[v,[u,w]]$ for homogeneous vectors in the indicated degrees. A Gerstenhaber algebra uses this sign rule after shifting degrees by one.

For a positively graded Lie algebra $L$ over $\mathbb{Q}$, finite-dimensional in each degree, its cochain algebra is free graded commutative on the graded dual of $sL$, with differential dual to the Lie bracket and its Koszul signs. The Jacobi identity gives $d^2=0$. A Lie element of degree $r$ contributes a cochain generator of degree $r+1$. Its cohomology computes ${\mathrm{Ext}}_{U(L)}(\mathbb{Q},\mathbb{Q})$ with total grading; the standard free enveloping-algebra resolution gives this identification. For a free graded Lie algebra, the tensor-algebra augmentation has a length-one free resolution, explaining the square-zero cohomology of a rational wedge sum of spheres.

For a positively graded vector space $V$, the free graded Lie algebra $\mathbb{L}(V)$ receives a linear map from $V$ and extends every graded linear map $V\to L$ uniquely to a Lie homomorphism. Over $\mathbb{Q}$, it is the Lie subalgebra generated by $V$ inside the tensor algebra $T(V)$, using $[u,v]=uv-(-1{)}^{|u||v|}vu$. Its universal enveloping algebra is $T(V)$: both have the same universal property for maps from $V$ into associative algebras. The primitives of the tensor Hopf algebra with primitive generators are exactly $\mathbb{L}(V)$.

The bracket on a graded Lie algebra is a bilinear operation satisfying graded antisymmetry and the graded Jacobi identity. For homogeneous elements of degrees $r,s$, $[u,v]=-(-1{)}^{rs}[v,u]$. A Gerstenhaber bracket uses this convention on the shifted degrees $r=|u|-1$ and $s=|v|-1$.

A central extension is a Lie algebra surjection whose kernel commutes with every element of the larger algebra. The Virasoro central extension adds a central generator to the Witt algebra.

Two central extensions of a Lie algebra $0\to \mathfrak{z}\to E\to \mathfrak{g}\to 0$ and $0\to \mathfrak{z}\to E^{\prime }\to \mathfrak{g}\to 0$ are equivalent if a Lie algebra isomorphism $E\to E^{\prime }$ commutes with both maps in the short exact sequences, inducing the identity on the specified $\mathfrak{z}$ and $\mathfrak{g}$. This is finer than asking only that $E$ and $E^{\prime }$ be isomorphic as Lie algebras.

For $\mathfrak{z}=kz$, choose linear sections and identify the two vector spaces with $\mathfrak{g}\oplus kz$. A linear map compatible with both ends necessarily has the form $(x,uz)\mapsto (x,(u+t(x))z)$ for a linear functional $t$. If the two Lie algebra two-cocycles are $\alpha ,{\alpha }^{\prime }$, preservation of the Lie bracket is exactly ${\alpha }^{\prime }(x,y)=\alpha (x,y)+t([x,y])$. Hence equivalence means that their difference is a Lie algebra two-coboundary. If changes of the identifications at the ends are allowed, an automorphism of a Lie algebra at either end can identify further classes. For a two-dimensional abelian Lie algebra and a one-dimensional central line, the finer classes are parametrized by $k$, whereas the middle Lie algebras have only the zero and nonzero classes, represented by the abelian Lie algebra $k^3$ and the Heisenberg Lie algebra.

The Witt algebra is the Lie algebra of vector fields on a circle with Laurent-polynomial coefficients. Its nontrivial central extension of a Lie algebra is the Virasoro algebra.

For an element $x$ of a Lie algebra $\mathfrak{g}$, its centralizer is the Lie subalgebra $Z_{\mathfrak{g}}(x)=\{y\in \mathfrak{g}:[y,x]=0\}$, equivalently the kernel of $\mathrm{ad}x$ in the Adjoint representation.

In a complex semisimple Lie algebra whose rank of a semisimple Lie algebra is $\ell$, a root vector $x$ satisfies $\dim Z_{\mathfrak{g}}(x)\ge 3\ell -2$. To see this, select $\ell -1$ roots whose images form a basis of the real root span modulo $\mathbb{R}\alpha$, where $x\in {\mathfrak{g}}_{\alpha }$. Take the highest endpoints of the root strings through both signs of each selected root. Their $2(\ell -1)$ distinct root spaces commute with $x$. Together with $\ker \alpha \subset \mathfrak{t}$ and $\mathbb{C}x$, these give $2(\ell -1)+(\ell -1)+1$ independent vectors in the Lie algebra centralizer.

A Lie algebra generator is an element of a chosen spanning or generating set for a Lie algebra. In a matrix representation, an infinitesimal group transformation has the form $1+i{\alpha }_a{t}^a+O({\alpha }^2)$ in terms of generator matrices $t^a$.

For a Matrix Lie group $G$, its tangent space $\mathfrak{g}=T_{I}G$ is closed under the matrix commutator. Indeed, for $X,Y\in \mathfrak{g}$, the group commutatorlies in $G$, and the mixed derivative at $(0,0)$ of its matrix logarithm is $XY-YX$. Thus $[X,Y]=XY-YX$ belongs to $\mathfrak{g}$.

If the matrix logarithm identifies a neighborhood of the identity in a Matrix Lie group with a neighborhood of zero in its tangent subspace, the logarithm of the displayed group commutator belongs to that subspace. Divide it by $t^2$ and take the limit. Finite-dimensional linear subspaces are closed, proving closure under the matrix commutator without attempting to give a nonsmooth square-root reparametrization a velocity.

The endomorphisms of a vector space form a Lie algebra under the commutator $[A,B]=AB-BA$. It is the Lie algebra of the general linear group; the Jacobi identity follows by expanding associative products.

The real Lie algebra of the unitary group is $\mathfrak{u}(N)=\{X:X^{†}=-X\}$, of real dimension $N^2$. Differentiating unitarity gives necessity; exponentiating any skew-Hermitian matrix gives a curve in the group, proving sufficiency.

The real Lie algebra of the special unitary group consists of traceless skew-Hermitian matrices. Differentiating unitarity and determinant one gives the two constraints; exponentiating any such matrix proves their sufficiency. The commutator is again skew-Hermitian and traceless. Its real dimension is $n^2-1$, and its complexification is ${\mathfrak{sl}}_n(\mathbb{C})$.

With the defining two-dimensional matrix trace and inner product $⟨X,Y⟩=-\mathrm{tr}(XY)$, the displayed Pauli matrix basis is orthonormal. The Pauli matrix multiplication law gives the indicated structure constants of a Lie algebra. With the alternative form $-2\mathrm{tr}(XY)$, the orthonormal basis is $-i{\sigma }_a/2$ and the structure constant is one. The bracket and inner-product normalizations must be kept consistent in projected gauge-field formulas.

The compact real Lie algebra $\mathfrak{su}(3)$ consists of traceless anti-Hermitian $3\times 3$ matrices. Physicists often specify Hermitian generators $t_a$ instead, so the genuine Lie algebra elements are $i{t}_a$. The compact real algebra and its complexification ${\mathfrak{sl}}_3(\mathbb{C})$ should be distinguished.

Each root $\alpha$ of the SU(3) Lie algebra defines an SU(2) Lie algebra generated by $J_3=\alpha \cdot H$ and $J_{\pm }=\sqrt{2}{E}_{\pm \alpha }$ in the root-length-one convention. They obey $[J_3,J_{\pm }]=\pm J_{\pm }$ and $[J_{+},J_{-}]=2J_3$. These subalgebras explain root strings, Weyl group reflections and the integer Dynkin labels of highest weights.

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A real basis of the unitary Lie algebra is $i{E}_{ii}$, $E_{ij}-E_{ji}$ and $i(E_{ij}+E_{ji})$ for $i<j$, where $E_{ij}$ are matrix units. These separately span the imaginary diagonal and real and imaginary upper off-diagonal entries.

A Lie algebra isomorphism is a bijective linear map $\phi :\mathfrak{g}\to \mathfrak{h}$ satisfying $\phi ([x,y])=[\phi (x),\phi (y)]$. Its inverse automatically preserves the Lie bracket.

The direct sum $\mathfrak{g}\oplus \mathfrak{h}$ has componentwise bracketThe two summands are commuting ideals.