Glossary of Lie algebras

This is a glossary for the terminology applied in the mathematical theories of Lie algebras. The statements in this glossary mainly focus on the algebraic sides of the concepts, without referring to Lie groups or other related subjects.

Definition

Lie algebra

A vector space

{\mathfrak {g}}

over a field

F

with a binary operation [·, ·] (called the Lie bracket or abbr. bracket) , which satisfies the following conditions:

\forall a,b\in F,x,y,z\in {\mathfrak {g}}

$[ax+by,z]=a[x,z]+b[y,z]$ (bilinearity)
$[x,x]=0$ (alternating)
$[[x,y],z]+[[y,z],x]+[[z,x],y]=0$ (Jacobi identity)

associative algebra: An associative algebra $A$ can be made to a Lie algebra by defining the bracket $[x,y]=xy-yx$ (the commutator of $x,y$ ) $\forall x,y\in A$ .

homomorphism: A vector space homomorphism $\phi :{\mathfrak {g}}_{1}\to {\mathfrak {g}}_{2}$ is said to be a Lie algebra homomorphism if $\phi ([x,y])=[\phi (x),\phi (y)]\,\forall x,y\in {\mathfrak {g}}_{1}.$

adjoint representation: Given $x\in {\mathfrak {g}}$ , define map ${\textrm {ad}}_{x}$ by; ${\begin{aligned}{\textrm {ad}}_{x}:&{\mathfrak {g}}\to {\mathfrak {g}}\\&y\mapsto [x,y]\end{aligned}}$; ${\textrm {ad}}_{x}$ is a Lie algebra derivation. The map; ${\begin{aligned}{\textrm {ad}}:&{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})\\&x\mapsto \mathrm {ad} _{x}\end{aligned}}$; thus defined is a Lie algebra homomorphism.; ${\textrm {ad}}:{\mathfrak {g}}\to {\textrm {End}}({\mathfrak {g}})$ is called adjoint representation.

Jacobi identity: The identity [[x, y], z] + [[y, z], x] + [[z, x], y] = 0.; To say Jacobi identity holds in a vector space is equivalent to say adjoint of all elements are derivations : ${\textrm {ad}}_{x}([y,z])=[{\textrm {ad}}_{x}(y),z]+[y,{\textrm {ad}}_{x}(z)]$ .

subalgebras

subalgebra: A subspace ${\mathfrak {g'}}$ of a Lie algebra ${\mathfrak {g}}$ is called the subalgebra of ${\mathfrak {g}}$ if it is closed under bracket, i.e. $[{\mathfrak {g'}},{\mathfrak {g'}}]\subseteq {\mathfrak {g'}}.$

ideal: A subspace ${\mathfrak {g'}}$ of a Lie algebra ${\mathfrak {g}}$ is the ideal of ${\mathfrak {g}}$ if $[{\mathfrak {g'}},{\mathfrak {g}}]\subseteq {\mathfrak {g'}}.$; In particular, every ideal is also a subalgebra. Every kernel of a Lie algebra homomorphism is an ideal. Unlike in ring theory, there is no distinguishability of left ideal and right ideal.

derived algebra: The derived algebra of a Lie algebra ${\mathfrak {g}}$ is $[{\mathfrak {g}},{\mathfrak {g}}]$ . It is a subalgebra.

normalizer: The normalizer of a subspace $K$ of a Lie algebra ${\mathfrak {g}}$ is $N_{\mathfrak {g}}(K):=\{x\in {\mathfrak {g}}|[x,K]\subseteq K\}$ .

centralizer: The centralizer of a subset $X$ of a Lie algebra ${\mathfrak {g}}$ is $C_{\mathfrak {g}}(X):=\{x\in {\mathfrak {g}}|[x,X]=\{0\}\}$ .

center: The center of a Lie algebra is the centralizer of itself : $Z(L):=\{x\in {\mathfrak {g}}|[x,{\mathfrak {g}}]=0\}$

radical: The radical ${\textrm {Rad}}({\mathfrak {g}})$ is the maximum solvable ideal of ${\mathfrak {g}}$ .

Solvability, nilpotency, Jordan decomposition, semisimplicity

abelian: A Lie algebra is said to be abelian if its derived algebra is zero.

nilpotent Lie algebra

A Lie algebra

L

is said to be nilpotent if

C^{N}(L)=\{0\}

for some positive integer

N

The following conditions are equivalent:

$C^{N}(L)=\{0\}$ for some positive integer $N$ , i.e. the descending central series eventually terminates to $\{0\}$ .
$C_{N}(L)=L$ for some positive integer N, i.e. the ascending central series eventually terminates to L.
There exists a chain of ideals of $L$ , $L=I_{1}\supseteq I_{2}\supseteq I_{3}\supseteq \cdots \supseteq I_{n}=\{0\}$ , such that $[L,I_{k}]\subseteq I_{k+1}$ .
There exists chain of ideals of $L$ , $L=I_{1}\supseteq I_{2}\supseteq I_{3}\cdots \supseteq I_{n}=\{0\}$ , such that $I_{k}/I_{k+1}\subseteq Z(L/I_{k+1})$ .
${\textrm {ad}}\,x$ is nilpotent $\forall x\in L$ . (Engel's theorem)
${\textrm {ad}}\,L$ is a nilpotent Lie algebra.

In particular, every nilpotent Lie algebra is solvable.

L

is nilpotent, any subalgebra and quotient of

L

are nilpotent.

nilpotent element in a Lie algebra: An element $x\in L$ is said to be nilpotent in $L$ if $ad_{x}$ is a nilpotent endomorphism, i.e. viewing ${\textrm {ad}}_{x}$ as a matrix in ${\mathfrak {gl}}_{\mathfrak {g}}$ , $\exists N\in \mathbb {Z} ^{+},({\textrm {ad}}_{x})^{N}=0$ . It is equivalent to $({\textrm {ad}}_{x})^{N}y=[x[x\ldots [x[x,y]\ldots ]=0\ \forall y\in L$

descending central series: a sequence of ideals of a Lie algebra $L$ defined by $C^{0}(L)=L,\,C^{1}(L)=[L,L],\,C^{n+1}(L)=[L,C^{n}(L)]$

ascending central series: a sequence of ideals of a Lie algebra $L$ defined by $C_{0}(L)=L,\,C_{1}(L)=Z(L)$ (center of L) , $C_{n+1}(L)=\pi _{n}^{-1}(Z(L/C_{n}(L)))$ , where $\pi _{i}$ is the natural homomorphism $L\to L/C_{n}(L)$

solvable Lie algebra: A Lie algebra $L$ is said to be solvable if $L^{(N)}=0$ for some positive integer $N$ , i.e. the derived series eventually terminates to $\{0\}$ .; The following condition is equivalent to solvability:; * There exists chain of ideals of $L$ , $L=I_{1}\supseteq I_{2}\supseteq I_{3}\cdots \supseteq I_{n}=\{0\}$ , such that $[I_{k},I_{k}]\subseteq I_{k+1}$ .

L

is solvable, any subalgebra and quotient of

L

are solvable.

Let

I

is an ideal of a Lie algebra

L

. If

L/I,I

are solvable,

L

is solvable.

derived series: a sequence of ideals of a Lie algebra L defined by $L^{(0)}=L,\,L^{(1)}=[L,L],\,L^{(n+1)}=[L^{(n)},L^{(n)}]$

simple: A Lie algebra is said to be simple if it is non-abelian and has only two ideals, itself and $\{0\}$ .

semisimple Lie algebra

A Lie algebra is said to be semisimple if its radical is

\{0\}

semisimple element in a Lie algebra

split Lie algebra

free Lie algebra

toral Lie algebra

Lie's theorem: Let ${\mathfrak {g}}$ be a finite-dimensional complex solvable Lie algebra over algebraically closed field of characteristic $0$ , and let $V$ be a nonzero finite dimensional representation of ${\mathfrak {g}}$ . Then there exists an element of $V$ which is a simultaneous eigenvector for all elements of ${\mathfrak {g}}$ .; Corollary: There exists a basis of $V$ with respect to which all elements of ${\mathfrak {g}}$ are upper triangular.

Killing form: The Killing form on a Lie algebra ${\mathfrak {g}}$ is a symmetric, associative, bilinear form defined by $\kappa (x,y):={\textrm {Tr}}({\textrm {ad}}\,x\,{\textrm {ad}}\,y)\ \forall x,y\in {\mathfrak {g}}$ .

Cartan criterion for solvability: A Lie algebra ${\mathfrak {g}}$ is solvable iff $\kappa ({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0$ .

Cartan criterion for semisimplity: If $\kappa (\cdot ,\cdot )$ is nondegenerate, then ${\mathfrak {g}}$ is semisimple.; If ${\mathfrak {g}}$ is semisimple and the underlying field $F$ has characteristic 0 , then $\kappa (\cdot ,\cdot )$ is nondegenerate.

lower central series: synonymous to "descending central series".

upper central series: synonymous to "ascending central series".

Root System (for classification of semisimple Lie algebra)

In the below section, denote $(\cdot ,\cdot )$ as the inner product of a Euclidean space E.

In the below section, $<\cdot ,\cdot >$ denoted the function defined as $<\beta ,\alpha >={\frac {(\beta ,\alpha )}{(\alpha ,\alpha )}}\,\forall \alpha ,\beta \in E$ .

Cartan subalgebra: A Cartan subalgebra ${\mathfrak {h}}$ of a Lie algebra ${\mathfrak {g}}$ is a nilpotent subalgebra satisfying $N_{\mathfrak {g}}({\mathfrak {h}})={\mathfrak {h}}$ .

regular element of a Lie algebra

maximal toral subalgebra

Borel subalgebra

root of a semisimple Lie algebra: Let ${\mathfrak {g}}$ be a semisimple Lie algebra, ${\mathfrak {h}}$ be a Cartan subalgebra of ${\mathfrak {g}}$ . For $\alpha \in {\mathfrak {h}}^{*}$ , let ${\mathfrak {g_{\alpha }}}:=\{x\in {\mathfrak {g}}|[h,x]=\alpha (h)x\,\forall h\in {\mathfrak {h}}\}$ . \alpha is called a root of ${\mathfrak {g}}$ if it is nonzero and ${\mathfrak {g_{\alpha }}}\neq \{0\}$; The set of all roots is denoted by $\Phi$ ; it forms a root system.

Root system

A subset

\Phi

of the Euclidean space

E

is called a root system if it satisfies the following conditions:

$\Phi$ is finite, ${\textrm {span}}(\Phi )=E$ and $0\notin \Phi$ .
For all $\alpha \in \Phi$ and $c\in \mathbb {R}$ , $c\alpha \in \Phi$ iff $c=\pm 1$ .
For all $\alpha ,\beta \in \Phi$ , $<\alpha ,\beta >$ is an integer.
For all $\alpha ,\beta \in \Phi$ , $S_{\alpha }(\beta )\in \Phi$ , where $S_{\alpha }$ is reflection through hyperplane normal to $\alpha$ i.e. $S_{\alpha }(x)=x-<x,\alpha >\alpha$ .

Cartan matrix: Cartan matrix of root system $\Phi$ is matrix $(<\alpha _{i},\alpha _{j}>)_{i,j=1}^{n}$ where $\Delta =\{\alpha _{1}\ldots \alpha _{n}\}$ is a set of simple roots of $\Phi$ .
Dynkin diagrams

Simple Roots

A subset

\Delta

of a root system

\Phi

is called a set of simple roots if it satisfies the following conditions:

$\Delta$ is linear basis of $E$ .
Each element of $\Phi$ is a linear combination of elements of $\Delta$ with coefficients which are either all nonnegative or all nonpositive.

a partial order on the Eucliean space E defined by the set of simple root: $\forall \lambda ,\mu \in E,\,\,\lambda >\mu \iff \lambda -\mu >0\iff \,\,\,\exists k_{1},k_{2},...,k_{n}\in \mathbb {Z} ^{+},\,\alpha _{1},\alpha _{2},...,\alpha _{n}\in \Delta ,\,\,\lambda -\mu =\sum _{i}k_{i}\alpha _{i}$

regular element with respect to a root system: Let $\Phi$ be a root system. $\gamma \in E$ is called regular if $(\gamma ,\alpha )\neq 0\,\forall \gamma \in \Phi$ .; For each set of simple roots $\Delta$ of $\Phi$ , there exists a regular element $\gamma \in E$ such that $(\gamma ,\alpha )>0\,\forall \gamma \in \Delta$ , conversely for each regular $\gamma$ there exist a unique set of base roots $\Delta (\gamma )$ such that the previous condition holds for $\Delta =\Delta (\gamma )$ . It can be determined in following way: let $\Phi ^{+}(\gamma )=\{\alpha \in \Phi |(\alpha ,\gamma )>0\}$ . Call an element $\alpha$ of $\Phi ^{+}(\gamma )$ decomposable if $\alpha =\alpha '+\alpha ''$ where $\alpha ',\alpha ''\in \Phi ^{+}(\gamma )$ , then $\Delta (\gamma )$ is the set of all indecomposable elements of $\Phi ^{+}(\gamma )$
positive roots: Positive root of root system $\Phi$ with respect to a set of simple roots $\Delta$ is a root of $\Phi$ which is a linear combination of elements of $\Delta$ with nonnegative coefficients.

negative roots: Negative root of root system $\Phi$ with respect to a set of simple roots $\Delta$ is a root of $\Phi$ which is a linear combination of elements of $\Delta$ with nonpositive coefficients.

long root

short root

Weyl group: Weyl group of a root system $\Phi$ is a (necessarily finite) group of orthogonal linear transformations of $E$ which is generated by reflections through hyperplanes normal to roots of $\Phi$

inverse of a root system: Given a root system $\Phi$ . Define $\alpha ^{v}={\frac {2\alpha }{(\alpha ,\alpha )}}$ , $\Phi ^{v}=\{\alpha ^{v}|\alpha \in \Phi \}$ is called the inverse of a root system.; $\Phi ^{v}$ is again a root system and have the identical Weyl group as $\Phi$ .

base of a root system: synonymous to "set of simple roots"

dual of a root system: synonymous to "inverse of a root system"

theory of weights

<--

weight in a root system: $\lambda \in E$ is called a weight if $<\lambda ,\alpha >\in \mathbb {Z} \,\forall \alpha \in \Phi$ . -->

weight lattice

weight space

dominant weight: A weight \lambda is dominant if $<\lambda ,\alpha >\in \mathbb {Z} ^{+}$ for some $\alpha \in \Phi$

fundamental dominant weight: Given a set of simple roots $\Delta =\{\alpha _{1},\alpha _{2},...,\alpha _{n}\}$ , it is a basis of $E$ . $\alpha _{1}^{v},\alpha _{2}^{v},...,\alpha _{n}^{v}\in \Phi ^{v}$ is a basis of $E$ too; the dual basis $\lambda _{1},\lambda _{2},...,\lambda _{n}$ defined by $(\lambda _{i},\alpha _{j}^{v})=\delta _{ij}$ , is called the fundamental dominant weights.

highest weight

minimal weight
multiplicity (of weight)

radical weight

strongly dominant weight

Representation theory

module: Define an action of ${\mathfrak {g}}$ on a vector space $V$ ( i.e. an operation ${\mathfrak {g}}\times V\to V,\,(x,v)\mapsto xv$ ) such that: $\forall a,b\in F,x,y\in {\mathfrak {g}},v,w\in V$ satisfy; # $(ax+by)v=a(xv)+b(yv)$; # $x(av+bw)=a(xv)+b(xw)$; # $[x,y]v=x(yv)-y(xv)$; Then $V$ is called a ${\mathfrak {g}}$ -module. (Remark: $V,{\mathfrak {g}}$ have the same underlying field $F$ .); Each ${\mathfrak {g}}$ -module corresponds to a representation ${\mathfrak {g}}\to {\mathfrak {gl}}_{V}$ .; A subspace W is a submodule (more precisely, sub ${\mathfrak {g}}$ -module) of $V$ if ${\mathfrak {g}}$ -module $W\subset V$ .

representation: For a vector space $V$ , if there is a Lie algebra homomorphism $\pi :{\mathfrak {g}}\to {\mathfrak {gl}}_{V}$ , then $\pi$ is called a representation of ${\mathfrak {g}}$ .; Each representation ${\mathfrak {g}}\to {\mathfrak {gl}}_{V}$ corresponds to a ${\mathfrak {g}}$ -module $V$ .; A subrepresentation is the representation corresponding to a submodule.

homomorphism: Given two ${\mathfrak {g}}$ -module V, W, a ${\mathfrak {g}}$ -module homomorphism $\phi$ is a vector space homomorphism satisfying $\phi (xv)=x\phi (v)\forall x\in {\mathfrak {g}},v\in V$ .

trivial representation: A representation is said to be trivial if the image of ${\mathfrak {g}}$ is the zero vector space. It corresponds to the action of ${\mathfrak {g}}$ on module $V$ by $xv=0\forall x\in {\mathfrak {g}},v\in V$ .

faithful representation: If the representation ${\mathfrak {g}}\to {\mathfrak {gl}}_{V}$ is injective, it is said to be faithful.

tautology representation: If a Lie algebra ${\mathfrak {g}}$ is defined as a subalgebra of ${\mathfrak {gl}}(n,F)$ , like ${\mathfrak {sl}}(n,F),{\mathfrak {o}}(2l,F),{\mathfrak {t}}(n,F)$ (the upper triangular matrices), the tautology representation is the imbedding ${\mathfrak {g}}\to {\mathfrak {gl}}(n,F)$ . It corresponds to the action on module $F^{n}$ by the matrix multiplication.

adjoint representation: The representation ${\begin{aligned}{\textrm {ad}}:&{\mathfrak {g}}\to {\textrm {gl}}_{\mathfrak {g}}\\&x\mapsto {\textrm {ad}}_{x}\end{aligned}}$ . It corresponds viewing ${\mathfrak {g}}$ as a ${\mathfrak {g}}$ -module - the action on the module is given by the adjoint endomorphism.

irreducible modules: A module is said to be irreducible if it has only two submodules, itself and zero.

indecomposable module: A module is said to be indecomposable if it cannot be written as direct sum of two non-zero submodules.; An irreducible module need not be indecmoposable but the converse is not true.

completely reducible module: A module is said to be completely reducible if it can be written as direct sum of irreducible modules.

simple module: Synonymous as irreduible module.

quotient module / quotient representation: Given a ${\mathfrak {g}}$ -module V and its submodule W, an action ${\mathfrak {g}}$ on V/W can be defined by $x(v+W)=xv+W\,\forall x\in {\mathfrak {g}},v\in V$ . V/W is said to be a quotient module in this case.

Schur's lemma: Statement in the language of module theory: Given V an irreducible ${\mathfrak {g}}$ -module, $\phi V\to V$ is a ${\mathfrak {g}}$ -module homomorphism iff $\phi =\lambda 1_{V}$ for some $\lambda \in F$ .; Statement in the language of representation theory: Given an irreducible representation $\phi \colon L\to {\mathfrak {gl}}(V)$ , for $\theta \in {\textrm {End}}(V)$ , $\theta \phi (x)=\phi (x)\theta$ iff $\theta =\lambda 1_{V}$ for some $\lambda \in F$ .

simple module: synonymous to "irreduible module".

factor module: synonymous to "quotient module".

Universal enveloping algebras

PBW theorem (Poincaré–Birkhoff–Witt theorem)

Verma modules

BGG category \mathcal{O}

Chevalley basis

a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups.

The generators of a Lie group are split into the generators H and E such that:

[H_{\alpha _{i}},H_{\alpha _{j}}]=0

[H_{\alpha _{i}},E_{\alpha _{j}}]=A_{ij}E_{\alpha _{j}}

[E_{\alpha _{i}},E_{\alpha _{j}}]=H_{\alpha _{j}}

[E_{\beta },E_{\gamma }]=\pm (p+1)E_{\beta +\gamma }

where p = m if β + γ is a root and m is the greatest positive integer such that γ − mβ is a root.

Examples of Lie algebra

general linear algebra $gl(n,F)$

Ado's theorem: Any finite-dimensional Lie algebra is isomorphic to a subalgebra of ${\mathfrak {gl}}_{V}$ for some finite-dimensional vector space V.

Simple Algebras

Classical Lie algebras:

Name	Root System	dimension	construction as subalgebra of ${\mathfrak {gl}}(n,F)$
Special linear algebra	$A_{l}\ (l\geq 1)$	$l^{2}+2l$	${\mathfrak {sl}}(l+1,F)=\{x\in {\mathfrak {gl}}(l+1,F)\|Tr(x)=0\}$ (traceless matrices)
Orthogonal algebra	$B_{l}\ (l\geq 1)$	$2l^{2}+l$	${\mathfrak {o}}(2l+1,F)=\{x\in {\mathfrak {gl}}(2l+1,F)\|sx=-x^{t}s,s={\begin{pmatrix}1&0&0\\0&0&I_{l}\\0&I_{l}&0\end{pmatrix}}\}$
Symplectic algebra	$C_{l}\ (l\geq 2)$	$2l^{2}-l$	${\mathfrak {sp}}(2l,F)=\{x\in {\mathfrak {gl}}(2l,F)\|sx=-x^{t}s,s={\begin{pmatrix}0&I_{l}\\-I_{l}&0\end{pmatrix}}\}$
Orthogonal algebra	$D_{l}(l\geq 1)$	$2l^{2}+l$	${\mathfrak {o}}(2l,F)=\{x\in {\mathfrak {gl}}(2l,F)\|sx=-x^{t}s,s={\begin{pmatrix}0&I_{l}\\I_{l}&0\end{pmatrix}}\}$

Exceptional Lie algebras:

Root System	dimension
G₂	14
F₄	52
E₆	78
E₇	133
E₈	248

Miscellaneous

Other discipline related

References

Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5

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