Lie theory

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 $G$.

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.

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.

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.

For a matrix Lie group, the left-invariant Maurer-Cartan form is $\rho =g^{-1}dg$. It identifies each tangent space with the Lie algebra and obeys $d\rho +\rho \wedge \rho =0$.

The Maurer-Cartan equation is $d\rho +\rho \wedge \rho =0$ for the left-invariant form $\rho =g^{-1}dg$.

If $\rho ={\sigma }^{\alpha }{T}_{\alpha }$ and $[T_{\alpha },T_{\beta }]=c^{\gamma }{}_{\alpha \beta }{T}_{\gamma }$, then

The orientation-preserving real affine group acts by $x\mapsto e^{a}x+b$ and has multiplication $(a,b)(a^{\prime },b^{\prime })=(a+a^{\prime },b+e^a{b}^{\prime })$.

A left-invariant Riemannian metric is preserved by every left translation. Its Killing fields generated by left translations are right-invariant vector fields.

For a matrix Lie group, the exponential map is the matrix exponential $\exp X={\sum }_{n\ge 0}{X}^n/n!$ and sends one-parameter additive subgroups of the Lie algebra to one-parameter subgroups of the group.

The Cayley transform is a rational local parametrization of a matrix Lie group from its Lie algebra wherever $I-X$ is invertible. Unlike the exponential map, its image can meet nonidentity components.

A Lie algebra over a field is a vector space $\mathfrak{g}$ with a bilinear operation $[x,y]$, called the Lie bracket, that is alternating and satisfies the Jacobi identity.

The Lie bracket is the product in a Lie algebra. For matrix Lie algebras it is the commutator $[X,Y]=XY-YX$.

The commutator of two elements of an associative algebra is $[A,B]=AB-BA$. Matrix Lie algebras use the commutator as their Lie bracket.

The Jacobi identity is

A Lie algebra is abelian when every Lie bracket vanishes.

A vector subspace $I\subseteq \mathfrak{g}$ is an ideal when $[\mathfrak{g},I]\subseteq I$. It is therefore the kernel of a Lie-algebra quotient map.

The derived series is ${\mathfrak{g}}^{(0)}=\mathfrak{g}$ and ${\mathfrak{g}}^{(n+1)}=[{\mathfrak{g}}^{(n)},{\mathfrak{g}}^{(n)}]$.

A Lie algebra is solvable when its derived series eventually becomes zero.

Every finite-dimensional complex representation of a solvable Lie algebra has a common eigenvector. Equivalently, an irreducible finite-dimensional complex representation is one-dimensional; iterating gives simultaneous upper triangularization.

A complex Lie subalgebra $\mathfrak{h}\subseteq \mathfrak{gl}(V)$ is solvable iffor every $x\in [\mathfrak{h},\mathfrak{h}]$ and $y\in \mathfrak{h}$. For an abstract Lie algebra, this is equivalent to $\kappa (\mathfrak{h},[\mathfrak{h},\mathfrak{h}])=0$.

The center is $Z(\mathfrak{g})=\{z:[z,x]=0\text{ for every }x\in \mathfrak{g}\}$.

The lower central series is defined by ${\gamma }_1(\mathfrak{g})=\mathfrak{g}$ and ${\gamma }_{i+1}(\mathfrak{g})=[\mathfrak{g},{\gamma }_i(\mathfrak{g})]$.

A Lie algebra is nilpotent when its lower central series eventually becomes zero.

A finite-dimensional Lie algebra is nilpotent if and only if every adjoint map ${\mathrm{ad}}_x$ is nilpotent. A Lie algebra of nilpotent endomorphisms can be simultaneously represented by strictly upper triangular matrices.

A Lie algebra homomorphism is a linear map $f$ satisfying $f([x,y])=[f(x),f(y)]$.

A representation of $\mathfrak{g}$ on a vector space $V$ is a Lie algebra homomorphism $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$.

For a basis $(T_a)$ of a Lie algebra, its structure constants are defined by $[T_a,T_b]=f_{ab}{}^c{T}_c$.

A Lie algebra representation is faithful when its representing homomorphism is injective.

A nonzero Lie algebra representation is irreducible when it has no proper nonzero invariant subspace.

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.

The Killing form isIt is symmetric and invariant: $\kappa ([X,Y],Z)=\kappa (X,[Y,Z])$.

The radical ${\mathfrak{g}}^{\perp }=\{x:\kappa (x,\mathfrak{g})=0\}$ of the Killing form is a solvable ideal. Invariance makes it an ideal, and the Cartan solvability criterion applied to its adjoint image proves solvability.

Every invariant bilinear form on a finite-dimensional complex simple Lie algebra is a scalar multiple of its Killing form. A nondegenerate invariant form identifies the algebra with its dual; comparing this identification with the Killing form gives an endomorphism of the irreducible adjoint representation, so Schur lemma makes it scalar.

For ${\mathfrak{sl}}_n(\mathbb{C})$,

Every invariant subspace of the Adjoint representation of a Lie algebra is an ideal. Hence the adjoint representation of a Simple Lie algebra is irreducible.

For a finite-dimensional Lie algebra representation $\rho :\mathfrak{g}\to \mathfrak{gl}(V)$, its trace form is the symmetric bilinear formIt is invariant: $([x,y],z{)}_V=(x,[y,z]{)}_V$.

For a finite-dimensional representation $d$, the trilinear form $B(X,Y,Z)=\mathrm{Tr}(d(X)d(Y)d(Z))$ is invariant under simultaneous adjoint action. This follows by writing the sum of its three infinitesimal variations as the trace of a commutator.

The three-dimensional Heisenberg Lie algebra has a basis $a,b,c$ with $[a,b]=c$ and $c$ central. It is a two-step Nilpotent Lie algebra.

On $\mathbb{C}[x]$, the operators $d/dx$, multiplication by $x$, and the identity satisfy $[d/dx,x]=1$. Sending $a,b,c$ to these operators gives a faithful irreducible infinite-dimensional representation of the Heisenberg Lie algebra.

The universal enveloping algebra $U(\mathfrak{g})$ is the associative algebra generated by $\mathfrak{g}$ subject to $xy-yx=[x,y]$. Its modules are the same as Lie algebra representations.

For an ordered basis $(x_i)$ of a Lie algebra $\mathfrak{g}$, the ordered monomials $x_1^{a_1}\cdots x_n^{a_n}$ form a basis of $U(\mathfrak{g})$. In particular, a triangular decomposition $\mathfrak{g}={\mathfrak{n}}^{-}\oplus \mathfrak{t}\oplus {\mathfrak{n}}^{+}$ gives $U(\mathfrak{g})\cong U({\mathfrak{n}}^{-})\otimes U(\mathfrak{t})\otimes U({\mathfrak{n}}^{+})$ as vector spaces.

A finite-dimensional Lie algebra is semisimple when it has no nonzero solvable ideals.

For a choice of positive roots, the corresponding Borel subalgebra is $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$, the sum of a Cartan subalgebra and all positive root spaces.

A nonabelian Lie algebra is simple when its only ideals are zero and the entire algebra.

The special linear Lie algebra ${\mathfrak{sl}}_n$ consists of the trace-zero $n\times n$ matrices with the commutator bracket.

The two fundamental representations of ${\mathfrak{sl}}_3$ are the defining representation $V({\omega }_1)={\mathbb{C}}^3$ and its dual

For the defining representation $V=V({\omega }_1)$ of ${\mathfrak{sl}}_3$,

The Lie algebra ${\mathfrak{sl}}_2$ has generators $e,h,f$ with $[h,e]=2e$, $[h,f]=-2f$, and $[e,f]=h$.

For every $n\ge 0$ there is one irreducible ${\mathfrak{sl}}_2(\mathbb{C})$-module $V(n)$ of dimension $n+1$, with weights $n,n-2,\dots ,-n$, each of multiplicity one. Every finite-dimensional representation is a direct sum of these modules.

On any finite-dimensional ${\mathfrak{sl}}_2$-module, the lowering operator $f$ is injective from the $m$-weight space to the $(m-2)$-weight space whenever $m>0$. This follows on each irreducible summand from its standard weight string.

The Verma module $M_{\lambda }$ has basis $(f^r{v}_{\lambda }{)}_{r\ge 0}$ andIt is reducible exactly when $\lambda =m\in {\mathbb{Z}}_{\ge 0}$; then its unique proper nonzero submodule is generated by $f^{m+1}{v}_{\lambda }$ and is isomorphic to $M_{-m-2}$.

For a nondegenerate alternating matrix $J$, the symplectic Lie algebra is ${\mathfrak{sp}}_{2n}=\{A:AJ+J{A}^T=0\}$.

In an orthonormal basis $({\epsilon }_i)$, the $C_n$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $i<j$ and $\pm 2{\epsilon }_i$.

The special orthogonal Lie algebra consists of the infinitesimal transformations preserving a nondegenerate symmetric bilinear form.

The Lorentz algebra has rotation generators $J_i$ and boost generators $K_i$ satisfying

Over the complex numbers, $A_i=(J_i+i{K}_i)/2$ and $B_i=(J_i-i{K}_i)/2$ generate commuting copies of ${\mathfrak{sl}}_2(\mathbb{C})$, givingFinite-dimensional irreducible representations are labelled $(j_L,j_R)$.

Parity fixes $J_i$, negates $K_i$, and therefore exchanges the two chiral factors. It sends $(j_L,j_R)$ to $(j_R,j_L)$.

The quadratic contraction $M_{\mu \nu }{M}^{\mu \nu }$ is parity even, while $M_{\mu \nu }{\tilde{M}}^{\mu \nu }$ is parity odd. Their linear combinations give the quadratic Casimirs of the two chiral ${\mathfrak{sl}}_2$ factors.

The Lie algebra ${\mathfrak{so}}_5$ has dimension ten and root system $B_2$.

Under the standard block-diagonal subgroup, the vector and adjoint representations branch asUsing ${\mathfrak{so}}_4\cong {\mathfrak{su}}_2\oplus {\mathfrak{su}}_2$, these are $(\frac{1}{2},\frac{1}{2})\oplus (0,0)$ and $(1,0)\oplus (0,1)\oplus (\frac{1}{2},\frac{1}{2})$.

The $B_2$ root system consists of $\pm {\epsilon }_1$, $\pm {\epsilon }_2$, and $\pm {\epsilon }_1\pm {\epsilon }_2$. Its vector representation has weights ${\epsilon }_1,{\epsilon }_2,0,-{\epsilon }_2,-{\epsilon }_1$.

The root systems $B_2$ and $C_2$ are isomorphic after interchanging long and short simple-root labels. The classification of complex simple Lie algebras therefore gives ${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$.

For simple roots $(1,0)$ and $(-1,1)$, the fundamental weights are ${\omega }_1=(1/2,1/2)$ and ${\omega }_2=(0,1)$. The representation $[1,0]$ is the four-dimensional spin representation with weights $(\pm 1/2,\pm 1/2)$; $[0,1]$ is the five-dimensional vector representation with weights $(\pm 1,0),(0,\pm 1),(0,0)$.

For a complex semisimple Lie algebra, a Cartan subalgebra is a maximal abelian subalgebra consisting of semisimple elements. It is also called a maximal torus in this setting.

For a Cartan subalgebra $\mathfrak{t}$, a semisimple Lie algebra decomposes asThe nonzero functionals $\alpha$ are the roots.

A root system is a finite set of nonzero vectors closed under the reflections they define and satisfying the crystallographic integrality condition when it arises from a semisimple Lie algebra.

A choice of positive roots selects exactly one of $\alpha$ and $-\alpha$ and is closed under addition whenever the sum is a root.

The simple roots are the positive roots that cannot be expressed as sums of two positive roots. Every root is an integer combination of simple roots with coefficients of one sign.

The highest root is the maximal positive root in the root order determined by the simple roots.

The Weyl vector is $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$. It satisfies $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ for every simple root.

The coroot associated with a nonzero root is ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha )$.

The root lattice is the integer span of the roots, equivalently of the simple roots.

For weights $\mu ,\lambda$, one writes $\mu ⪯\lambda$ when $\lambda -\mu$ is a nonnegative integer combination of the simple roots.

The weight lattice is the set of vectors $\lambda$ for which $⟨\lambda ,{\alpha }^{\vee }⟩$ is an integer for every root $\alpha$.

An integral weight $\lambda$ is dominant when $⟨\lambda ,{\alpha }_i^{\vee }⟩\ge 0$ for every simple coroot. Equivalently, it is a nonnegative integer combination of the fundamental weights.

The fundamental weights are dual to the simple coroots: $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$.

The fundamental representation associated with a fundamental weight ${\omega }_i$ is the irreducible highest-weight representation $V({\omega }_i)$.

The reflection associated with a root is $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$.

The reflection group of a root system $\Phi$ is the subgroup of the orthogonal group generated by the reflections $s_{\alpha }$ for $\alpha \in \Phi$. It permutes the roots and acts freely and transitively on the fundamental systems.

The Weyl group is generated by the reflections in the roots.

The Coxeter length $\ell (w)$ is the smallest number of simple reflections whose product is $w$. Its parity gives the sign of a Weyl-group element.

If $\lambda$ and $w\lambda$ both lie in the closed dominant chamber, then $w$ is a product of simple reflections whose walls contain $\lambda$. In particular, $w\lambda =\lambda$.