Algebra

A normed algebra is an algebra over a field with a norm satisfying $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$. Its completion is a Banach algebra, since multiplication extends continuously. Completeness is additional structure, rather than part of the definition of a normed algebra.

A finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

Suppose a ring has an underlying real normed vector space and its norm is multiplicative, without initially assuming real bilinearity of multiplication. For fixed $x$, distributivity makes $y\mapsto xy$ additive, while $\Vert xy-xz\Vert =\Vert x\Vert \Vert y-z\Vert$ makes it continuous. Additivity gives rational scalar compatibility, and continuity extends it to every real scalar. Apply the same argument in the other variable to obtain a bilinear map. The norm identity also excludes nonzero zero divisors.

A normed division algebra is a nonzero unital normed algebra in which every nonzero element has a two-sided multiplicative inverse. The complex Gelfand-Mazur theorem forces such an algebra to be $\mathbb{C}$, even when completeness is not initially assumed: embed it in its Banach algebra completion and use nonemptiness of the Banach-algebra spectrum.

For an additive semigroup $D$ with identity zero, the semigroup algebra $k[D]$ has a $k$-basis of formal symbols $t^q$, $q\in D$, and multiplication $t^q{t}^{q^{\prime }}=t^{q+q^{\prime }}$. For $D=\mathbb{Z}[1/2{]}_{\ge 0}$ it is the integral domain ${\bigcup }_{h\ge 0}k[t^{1/2^h}]$. Its positive-exponent ideal is a nonzero idempotent ideal, and the strictly ascending chain $(t)⊊(t^{1/2})⊊\cdots$ shows that the algebra is not a Noetherian ring.

A semiring has a commutative monoid operation $(S,+,0)$ and a monoid operation $(S,\cdot ,1)$, with multiplication distributing over addition on both sides and $0$ absorbing multiplication. Additive inverses are not required. The natural numbers with their usual addition and multiplication are a basic example. Regarding multiplication as composition yields a one-object category with commutative-monoid enrichment.

Bernoulli's inequality states that $(1+x{)}^r\ge 1+rx$ for every integer $r\ge 0$ and real $x\ge -1$. In particular, $(1-x{)}^r\ge 1-rx$ for $0\le x\le 1$.

An algebraic operation combines one or more elements of a set to produce another element.

A binary operation on a set $S$ is a function $S\times S\to S$.

A binary operation is commutative if $a\ast b=b\ast a$ for every pair in its domain. A particular pair may commute even when the operation is not commutative. For example, ordinal addition and ordinal multiplication are not generally commutative.

A binary operation $\ast$ is associative when $(x\ast y)\ast z=x\ast (y\ast z)$ for every $x,y,z$.

A semigroup is a set equipped with an associative operation.

An element of a semigroup is idempotent when multiplying it by itself leaves it unchanged. Every element of a left-zero or right-zero semigroup is idempotent, whereas a group has only its identity as an idempotent. The Ellis–Numakura lemma guarantees such an element in a nonempty compact one-sided topological semigroup. Additive idempotence on ultrafilters gives the idempotent ultrafilter relation $p+p=p$.

A semigroup presentation specifies generators $X$ and equalities between nonempty words. The resulting semigroup is the quotient of the free semigroup $X^{+}$ by the smallest congruence containing these equalities. Two words are equal precisely when one can be transformed into the other by finitely many replacements in contexts, using the defining equalities in either direction.

A semigroup presentation is finite when both its generator set and its list of defining equalities are finite. A finite transition table for a Turing machine can be encoded this way: state-and-tape words simulate transitions, boundary rules supply blank cells, and halting-state rules erase the configuration to one distinguished symbol.

A subsemigroup is a nonempty subset of a semigroup closed under its associative operation. In a compact Hausdorff space carrying a left-topological semigroup, descending chains of nonempty compact subsemigroups have a nonempty compact subsemigroup as their intersection. This supplies the minimal object used in the Ellis–Numakura lemma.

A monoid is a semigroup with a two-sided identity element.

A left monoid action assigns each $m\in M$ a map $a\mapsto m\cdot a$ with $1\cdot a=a$ and $(mn)\cdot a=m\cdot (n\cdot a)$. It need not be invertible or injective. A left M-set is equivalently a covariant functor from the one-object category determined by $M$.

A right monoid action has $x1=x$ and $(xm)n=x(mn)$. It is a contravariant functor from the one-object category of the monoid to sets. The regular right action is multiplication on $M$ itself. Products of right actions use the diagonal action.

Give $M\times X$ the diagonal right monoid action. On its equivariant maps into $Y$, define $(ek)(g,x)=e(kg,x)$. The evaluation map of an exponential object is $e(1,x)$ and currying of $h:Z\times X\to Y$ is $\hat{h}(z)(g,x)=h(zg,x)$. Both maps are equivariant, and they are inverse under evaluation. The left multiplication $kg$ in this right action is essential in a noncommutative monoid.

For a group, evaluation at the identity identifies the monoid exponential with all set functions $X\to Y$. Its inverse is $e_f(g,x)=f(x{g}^{-1})g$, and the resulting function action is the displayed formula. Fixed functions are precisely equivariant maps. The inverse on the input is necessary for equivariance of evaluation and for the right-action law.

An equivariant map $h:A\to B$ between left M-sets obeys $h(m\cdot a)=m\cdot h(a)$. These are the natural transformations of the corresponding functors from the one-object monoid category.

An M-set is a set carrying a monoid action. This terminology here uses left actions. Morphisms are equivariant maps of monoid sets, and the resulting functor category $[M,\mathbf{Set}]$ is an elementary topos.

A monoid homomorphism $h:M\to N$ preserves multiplication and identity: $h(xy)=h(x)h(y)$ and $h(1_M)=1_N$. It therefore preserves every finite ordered product, including the empty product. The monoids with these maps form the category $\mathbf{Mon}$.

A monoid automorphism is a bijective operation- and identity-preserving self-map of a monoid. Its inverse is also a monoid homomorphism. Permuting prime numbers and extending by prime factorization gives all automorphisms of the multiplicative monoid of positive natural numbers; extending by $0\mapsto 0$ gives automorphisms of the multiplicative monoid including zero.

The free monoid on an alphabet consists of all finite words in its letters, including the empty word, under concatenation. Distinct words represent distinct elements.

A commutative monoid is a monoid whose operation is commutative. The natural numbers with addition form a commutative monoid that is not an abelian group, because positive elements have no additive inverses.

A commutative monoid is cancellative when $a+c=b+c$ implies $a=b$. Cancellation makes the natural map into its Grothendieck group injective.

A cancellative commutative monoid is torsion-free in this sense when $na=nb$ implies $a=b$ for every positive integer $n$. Its Grothendieck group is then a torsion-free abelian group, because $n(a-b)=0$ forces $na=nb$. This is the condition inherited by a submonoid of a torsion-free abelian group.

This category has commutative monoids and their homomorphisms. Pointwise addition makes each hom-set a commutative monoid, and composition is additive in both variables. Finite cartesian products are also coproducts, using the coordinate injections with zero in the other coordinates and summing images. Therefore it is a semi-additive category, but it is not an additive category. The reflexive pair from the submonoid $\{(m,n):m\le n\le 2m\}$ of ${\mathbb{N}}^2$ to $\mathbb{N}$, using the two projections and section $m\mapsto (m,m)$, has an arrow from $1$ to $2$ and none from $2$ to $1$ in its represented graph. Thus a reflexive pair here need not yield a groupoid.

An additive semigroup is a semigroup whose operation is written as addition.

A topological semigroup is a semigroup with a topology for which multiplication is jointly continuous.

A left-topological semigroup requires only that every right translation $x\mapsto xa$ be continuous.

Every nonempty compact Hausdorff left-topological semigroup contains an idempotent. Choose a minimal nonempty compact subsemigroup $K$. For $a\in K$, minimality gives $Ka=K$; the nonempty compact subsemigroup $\{x\in K:xa=a\}$ is therefore all of $K$, so $a^2=a$.

A matrix is strictly diagonally dominant when $|a_{ii}|>{\sum }_{j\ne i}|a_{ij}|$ in every row.

Lie theory studies continuous symmetries through Lie groups, Lie algebras, and their representations.

An affine algebraic group is an affine variety $G$ whose multiplication and inversion are regular maps. Equivalently, its coordinate ring $k[G]$ is a commutative Hopf algebra, with comultiplication induced by multiplication in $G$.

For a finite-dimensional vector space $V$, the affine group consists of invertible affine maps $x\mapsto Ax+b$, where $A$ belongs to the general linear group and $b\in V$. Its product is $(A,b)(A^{\prime },b^{\prime })=(A{A}^{\prime },b+A{b}^{\prime })$, making it the semidirect product of the translation vector space and its general linear group. On the real line the positive-dilation subgroup is the real affine group used in the adjacent article.

A rational representation is a group homomorphism whose matrix coefficients are regular functions on the affine algebraic group. For the general linear group, those functions are polynomials in matrix entries and the inverse determinant. The word rational does not permit poles at points of the group.

For finite-dimensional rational representations of complex $G{L}_m$ or $S{L}_m$, average a Hermitian inner product over the compact group $U(m)$ or $SU(m)$ using normalized Haar measure. Orthogonal complements become invariant under the compact group and its complexified Lie algebra. The latter generates the complex group, giving invariant complements and hence complete reducibility. This conclusion does not hold for arbitrary affine algebraic groups, such as the additive group.

The matrix coefficients of a rational $G{L}_m$ action belong to its coordinate ring $\mathbb{C}[g_{ab},\det (g{)}^{-1}]$. Multiplying the action by a sufficiently large determinant twist clears all determinant denominators and produces a polynomial representation. Irreducible rational actions are indexed by decreasing integer highest weights, with negative coordinates allowed.

For a dominant integer tuple $\lambda$, choose $k=\max (0,-{\lambda }_m)$ and define $D_{\lambda }={\det }^{-k}\otimes D_{\lambda +k(1,\dots ,1)}$, using the polynomial Schur module on the right. The alternant character formula for the general linear group shows that a larger admissible shift yields the same irreducible representation.

The dual of $D_{\lambda }$ has dominant label $-w_0\lambda =(-{\lambda }_m,\dots ,-{\lambda }_1)$. Substitution $x_i\mapsto x_i^{-1}$ in the alternant character formula for the general linear group and reversal of determinant columns gives that character. The numerator and denominator reversal signs and common monomial factors cancel.

The irreducible rational $G{L}_m$ modules are $D_{\lambda }(V)=(\det V{)}^{{\lambda }_m}\otimes D_{\lambda -{\lambda }_m(1,\dots ,1)}(V)$. The second factor is a Schur module. Clearing determinant denominators reduces completeness to the Schur algebra classification of polynomial modules. Distinct dominant integer tuples have distinct highest torus weights. Their characters are symmetric Laurent polynomials; only polynomial modules have characters defined at every singular endomorphism.

Split a rational $S{L}_m$ representation by its finite scalar center: on $W_k$, $\rho (\zeta I)={\zeta }^{k}I$ for ${\zeta }^m=1$. The displayed extension is independent of the scalar-root choice and is a group homomorphism. A matrix coefficient of $\rho {|}_{W_k}$ can be represented on $S{L}_m$ by a sum of homogeneous polynomials $P_d$ with $d\equiv k(\mathrm{mod}m)$, by averaging over the scalar center. Its extension is ${\sum }_d(\det g{)}^{(k-d)/m}{P}_d(g)$, a regular function on $G{L}_m$. Every invariant subspace remains invariant under this extension, so irreducibility is preserved in both directions.

Tensoring a rational representation of the general linear group with a determinant power shifts every highest-weight coordinate by the same integer $r$. Positive twists can clear determinant denominators; negative twists supply rational representations that do not extend to singular matrices. Twisting preserves irreducibility and dimension.

Every one-dimensional rational $G{L}_m$ representation is ${\det }^r$ for one integer $r$. Its restriction to the diagonal torus is a Laurent monomial. Invariance under conjugation by permutation matrices forces all its exponents to agree, and density of diagonalizable invertible matrices gives the result on the whole group. For $G{L}_1$, the character identity $p(zw)=p(z)p(w)$ directly forces a Laurent polynomial to be one monomial with coefficient one.

In a polynomial representation, every matrix coefficient is a polynomial in the matrix entries. The action extends to the monoid of all matrices. The scalar action splits it into homogeneous polynomial degrees; each homogeneous degree-$n$ category is the module category of the Schur algebra $S(m,n)$. Its irreducibles are the Schur modules $D_{\lambda }({\mathbb{C}}^m)$ with $\lambda ⊢n$ and at most $m$ rows.

The symmetric algebra on $V\oplus {\Lambda }^{2}V$ contains every irreducible polynomial representation of $GL(V)$ exactly once. Its torus character is ${\prod }_i(1-x_i{)}^{-1}{\prod }_{i<j}(1-x_i{x}_j{)}^{-1}$, which equals ${\sum }_{\lambda }{s}_{\lambda }(x)$ by the Schur identity. Complete reducibility and linear independence of Schur characters identify the summands. This is a formal graded identity: each fixed scalar degree is finite, so no analytic convergence is needed. Multiplicity one refers to irreducible modules, not arbitrary reducible representations.

For a partition of an integer $\lambda ⊢n$, applying a Young symmetrizer of that shape to the tensor power of each vector space gives a Schur functor. The construction respects linear maps because their tensor powers commute with place permutations. For single rows it gives symmetric powers, and for single columns exterior powers.

A Schur module is the value of a Schur functor on a vector space. Over the complex numbers it is either zero or an irreducible polynomial representation of the general linear group. Its character is the Schur polynomial $s_{\lambda }$ in the eigenvalues, and its highest weight is $\lambda$ padded with zeros.

A column of length greater than $m$ antisymmetrizes more than $m$ vectors, giving zero. When there are at most $m$ rows, place the $i$th basis vector in every tensor position of row $i$. Row symmetrization multiplies by a nonzero factorial product, and column antisymmetrization is nonzero because the vectors in every column are distinct. This proves the precise nonvanishing criterion for a Schur module.

The Schur algebra is the commutant of place permutations on a tensor power. It is the linear span of the diagonal general linear group action, by Schur–Weyl duality. Modules over $S(m,n)$ correspond to homogeneous degree-$n$ polynomial representations. Over the complex numbers it is a semisimple algebra: decompose the tensor power as a module for the semisimple group algebra $\mathbb{C}{S}_n$, and take its endomorphism algebra.

The commutant of the permutation action on $V^{\otimes n}$ identifies with the symmetric tensors in $\mathrm{End}(V{)}^{\otimes n}$. Polarization spanning of symmetric tensors spans it by $a^{\otimes n}$. Interpolating $(a+tI{)}^{\otimes n}$ at $n+1$ values where $a+tI$ is invertible expresses each such power using invertible ones. Hence the Schur algebra is the linear span of the diagonal general linear action.

The commuting actions of the symmetric group and the general linear group on a tensor power are mutual commutants. The displayed sum ranges over partitions with at most $\dim V$ rows. The Specht modules and Schur modules are the simple factors for the two actions. In particular a primitive Young symmetrizer selects one copy of the matching Schur module.

If $\sigma$ has ${\alpha }_r$ cycles of length $r$, contraction of matrix entries around its cycles gives $\mathrm{tr}(\sigma {\xi }^{\otimes n})={\prod }_r\mathrm{tr}({\xi }^r{)}^{{\alpha }_r}$. The formula holds for every endomorphism, not only diagonalizable ones. The Schur–Weyl duality decomposition equates it with a sum of products of Specht and Schur characters.

The standard-tableau right-ideal decomposition of $\mathbb{C}{S}_n$, tensored over that algebra with $V^{\otimes n}$, gives this direct sum of general linear group modules. With $e_t=h_t/H_{\lambda }$, the map $e_t\mathbb{C}{S}_n\otimes T\to e_{t}T$ is an isomorphism. Individual summands generally need not be symmetric group submodules; instead each is a Schur module for the commuting action.

The Lie algebra of an affine algebraic group $G$ is the tangent space $T_{e}G$ at its identity. Its bracket is the commutator of the corresponding left-invariant derivations of the coordinate ring.

The irreducible rational representations of $S{L}_2(\mathbb{C})$ are ${\mathrm{Sym}}^n({\mathbb{C}}^2)$. Such a representation descends through $S{L}_2\to PG{L}_2$ exactly when the central element $-I$ acts trivially, which holds exactly when $n$ is even.

Every affine algebraic group has a faithful finite-dimensional rational group representation. Choose algebra generators of $k[G]$ and place them in a finite-dimensional subspace stable under right translations. An element acting trivially on that subspace has the same coordinate values as the identity and therefore is the identity.

The group $G\times G$ acts on $k[G]$ byFor $G={\mathbb{G}}_m$, this gives $k[G]={⨁}_{r\in \mathbb{Z}}{k}_{(-r,r)}$. For $G={\mathbb{G}}_a$, the action factors through $(a,b)\mapsto b-a$ and the degree filtration of $k[t]$ has trivial one-dimensional successive quotients.

An element of an affine algebraic group is semisimple when its image in one, equivalently every, faithful finite-dimensional representation is a diagonalizable linear map.

An element of an affine algebraic group is unipotent when its image in one, equivalently every, faithful finite-dimensional representation has every eigenvalue equal to one.

A square matrix $U$ is unipotent when $U-I$ is nilpotent. Over an algebraically closed field, this is equivalent to all eigenvalues being one. If $j$ belongs to a nilpotent ideal of an operator algebra, $I+j$ is a unipotent matrix. This realizes the radical factor in the Levi decomposition of a quiver automorphism group.

Every element $g$ of an affine algebraic group has unique commuting semisimple and unipotent parts $g_s,g_u$ such that $g=g_s{g}_u$. The construction agrees with the multiplicative Jordan decomposition in every rational representation.

The derived subgroup $[G,G]$ is the closed algebraic subgroup generated by the commutators $xy{x}^{-1}{y}^{-1}$. If $G$ is connected, images of finite products of commutators are connected and their closures stabilize by dimension, which shows that $[G,G]$ is connected.

An affine algebraic group is solvable when its derived series reaches the identity. A connected solvable affine algebraic group can be conjugated into an upper triangular matrix group by the Lie-Kolchin theorem.

Every connected solvable affine algebraic subgroup of $\mathrm{GL}(V)$ preserves a complete flag in $V$, equivalently it is conjugate to a subgroup of the upper triangular matrices.

A unipotent algebraic group is an affine algebraic group all of whose elements are unipotent. The Kolchin theorem embeds it into an upper unitriangular group.

It is the largest connected normal unipotent subgroup of a linear algebraic group. A connected reductive algebraic group has trivial unipotent radical. The subgroup $1+J(E)$ of a finite-dimensional algebra's unit group is closed, connected and unipotent.

Every unipotent algebraic subgroup of ${\mathrm{GL}}_n$ is conjugate to a subgroup of the upper unitriangular group. The filtration by vanishing initial superdiagonals then proves that every unipotent algebraic group is a nilpotent group.

A diagonalizable algebraic group is isomorphic to a closed subgroup of a product of copies of ${\mathbb{G}}_m$. Every rational representation of such a group decomposes into weight spaces.

A reductive algebraic group is a smooth connected affine algebraic group whose largest connected normal unipotent algebraic group is trivial.

A Borel subgroup is a maximal closed connected solvable subgroup of an affine algebraic group.

A parabolic subgroup of a connected reductive group is a closed subgroup containing a Borel subgroup. Its quotient in the group is projective.

A Levi subgroup is a reductive complement to the unipotent radical of a parabolic subgroup.

For a connected reductive algebraic group $G$ and a Borel subgroup $B$, the quotient $G/B$ is its complete flag variety. Quotients $G/P$ by parabolic subgroups are partial flag varieties.

A root datum is a quadruple $(X^{\ast },\Phi ,X_{\ast },{\Phi }^{\vee })$ of dual lattices, roots, and coroots with the natural perfect pairing and reflection axioms. A reductive group with maximal torus $T$ has $X^{\ast }=X^{\ast }(T)$ and $X_{\ast }=X_{\ast }(T)$.

For a connected reductive algebraic group, a Borel subgroup $B$, and a maximal torus $T\subseteq B$, the Weyl group $W=N_G(T)/T$ indexes the double cosets:

For an algebraic group $H$, a flat $H$-torsor over $Y$ is a faithfully flat morphism $X\to Y$ with a right $H$-action such that $X\times H\to X{\times }_{Y}X$, $(x,h)\mapsto (x,xh)$, is an isomorphism. A Zariski torsor additionally becomes $U\times H$ on a Zariski open cover of $Y$.

A Lie group is a group that is also a smooth manifold, with smooth multiplication and inversion. Its tangent space at the identity carries a natural Lie algebra structure.

This Lie group acts on the plane by $v\mapsto x+\rho v$ with $\rho >0$. Its multiplication is $(\rho ,x)({\rho }^{\prime },x^{\prime })=(\rho {\rho }^{\prime },x+\rho x^{\prime })$, and its inverse is $({\rho }^{-1},-{\rho }^{-1}x)$. It is a semidirect product in which positive dilation acts on the translation subgroup. A dilation generator $D$ and translation generators $T_1,T_2$ have Lie brackets $[D,T_a]=T_a$ and $[T_1,T_2]=0$.

The Maurer-Cartan form of the translation-dilation group of the plane has these coefficients in its dilation–translation basis, with $a=1,2$. Left multiplication rescales each coordinate differential and $\rho$ equally, proving they are left-invariant differential forms. Their dual left-invariant vector fields are $\rho {\partial }_{\rho }$ and $\rho {\partial }_{x^a}$. The Maurer-Cartan equation is $d{\lambda }^0=0$, $d{\lambda }^a=-{\lambda }^0\wedge {\lambda }^a$. The tensor ${\sum }_j{\lambda }^j\otimes {\lambda }^j$ is a positive definite left-invariant metric.

A smooth homomorphism of Lie groups has differential $D_{e}F$ at the identity. Differentiating $F\circ L_g=L_{F(g)}\circ F$ makes the corresponding left-invariant vector fields $F$-related. Their commutator action on pulled-back functions shows that the differential of a Lie group homomorphism preserves Lie brackets. The map need not be injective or surjective.

A Riemannian metric on a Lie group is right-invariant when every right translation is an isometry. It is determined by a positive-definite inner product at the identity, extended with a coframe of right-invariant differential forms. Its translation-generated Killing vector fields are left-invariant, rather than the right-invariant vector fields dual to that coframe.

A differential form $\alpha$ on a Lie group is right-invariant if $R_g^{\ast }\alpha =\alpha$ for every right translation $R_g(p)=pg$. A right Maurer-Cartan form is represented in a matrix group by $dg{g}^{-1}$, whereas the left version is $g^{-1}dg$. Its components supply a right-invariant coframe.

A maximal torus is a connected compact abelian Lie subgroup maximal among such subgroups. Restricting a finite-dimensional unitary representation to it splits into one-dimensional weight characters. For special unitary group $\mathrm{SU}(2)$ it is the diagonal circle $\mathrm{diag}(z,z^{-1})$.

The connected component of the identity in a Lie group is an open normal subgroup. Connected coordinate neighbourhoods prove openness, and conjugation preserves the component of the identity. Every open identity neighbourhood generates it: the generated subgroup is open, its other cosets are open, and connectedness forces it to be the whole component.

The group manifold is the underlying smooth manifold of a Lie group, with smooth multiplication and inversion. Its global topology need not be determined by its Lie algebra: the SU(2) group and SO(3) group have isomorphic Lie algebras but different fundamental groups.

A Lie subgroup is a subgroup carrying a Lie group structure such that its inclusion is an injective immersion and a group homomorphism. Its Lie algebra includes in that of the ambient Lie group.

Conjugation by every element of $G$ preserves a normal Lie subgroup $H$. Its differential therefore preserves $\mathfrak{h}=T_{e}H$. Differentiate that invariance along $\exp (tX)$ to obtain $\mathrm{ad}(X)\mathfrak{h}\subseteq \mathfrak{h}$, proving that $\mathfrak{h}$ is an ideal of a Lie algebra. Closed subgroups have a canonical Lie-subgroup structure; an arbitrary abstract subgroup need not.

A Lie group action is a smooth group action of a Lie group on a smooth manifold. Differentiating the action gives an infinitesimal action of its Lie algebra by vector fields.

A smooth action of the compact abelian Lie group ${\mathbb{R}}^n/(2\pi \mathbb{Z}{)}^n$. Commuting vector fields with a common periodic normalization give such an action. In action-angle variables, the torus acts by translation of the angles while leaving the actions fixed.

The special linear group acts on determinant-one real symmetric matrices by matrix congruence. Symmetry and determinant are preserved, and signatures are preserved by Sylvester's law of inertia. The determinant-one set is a regular level set of dimension $n(n+1)/2-1$, with tangent directions $C=C^T$ satisfying $\mathrm{tr}(B^{-1}C)=0$. Its infinitesimal left-action fields are $XB+B{X}^T$; converting them to a Lie algebra representation requires the infinitesimal left-action sign convention.

The coadjoint action of a Lie group on the dual space of its Lie algebra is $({\mathrm{Ad}}_g^{\ast }\eta )(X)=\eta ({\mathrm{Ad}}_{g^{-1}}X)$. The inverse in this formula makes it a left group action.

For a Lie group action and $X$ in its Lie algebra, the fundamental vector field is $X_M(p)={\frac{d}{dt}|}_{t=0}\exp (tX)\cdot p$.

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.

A matrix Lie group is a subgroup of a general linear group that is also a Lie group for its matrix topology. Near the identity, the matrix exponential and matrix logarithm give inverse coordinate charts whenever they are restricted to sufficiently small neighborhoods in the group and its tangent space.

For a real quadratic form $Q(v)=v^{T}Av$ with $A$ a symmetric matrix, its stabilizer is $G_Q=\{g\in {\mathrm{GL}}_n(\mathbb{R}):g^{T}Ag=A\}$. Its Lie algebra consists exactly of $X$ satisfying $X^{T}A+AX=0$. Differentiation proves necessity; the matrix exponential proves sufficiency because $e^{t{X}^T}A{e}^{tX}=A$. The result holds even when the quadratic form is degenerate.