Glossary of Lie groups and Lie algebras

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
v t e

Notations:

Throughout the glossary, ${\displaystyle (\cdot ,\cdot )}$ denotes the inner product of a Euclidean space E and ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes the rescaled inner product

${\displaystyle \langle \beta ,\alpha \rangle ={\frac {(\beta ,\alpha )}{(\alpha ,\alpha )}}\,\forall \alpha ,\beta \in E.}$

abelian

1.  An abelian Lie group is a Lie group that is an abelian group.

2.  An abelian Lie algebra is a Lie algebra such that ${\displaystyle [x,y]=0}$ for every ${\displaystyle x,y}$ in the algebra.

adjoint

1.  An adjoint representation of a Lie group:

${\displaystyle \operatorname {Ad} :G\to \operatorname {GL} ({\mathfrak {g}})}$

such that ${\displaystyle \operatorname {Ad} (g)}$ is the differential at the identity element of the conjugation ${\displaystyle c_{g}:G\to G,x\mapsto gxg^{-1}}$.

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

B

1.  (B, N) pair

Borel

1.  Armand Borel (1923 – 2003), a Swiss mathematician

2.  A Borel subgroup.

3.  A Borel subalgebra is a maximal solvable subalgebra.

4.  Borel-Bott-Weil theorem

Bruhat

1.  Bruhat decomposition

Cartan

1.  Élie Cartan (1869 – 1951), a French mathematician

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

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

4.  Cartan criterion for semisimplicity: (1) If ${\displaystyle \kappa (\cdot ,\cdot )}$ is nondegenerate, then ${\displaystyle {\mathfrak {g}}}$ is semisimple. (2) If ${\displaystyle {\mathfrak {g}}}$ is semisimple and the underlying field ${\displaystyle F}$ has characteristic 0 , then ${\displaystyle \kappa (\cdot ,\cdot )}$ is nondegenerate.

5.  The Cartan matrix of the root system ${\displaystyle \Phi }$ is the matrix ${\displaystyle (\langle \alpha _{i},\alpha _{j}\rangle )_{i,j=1}^{n}}$, where ${\displaystyle \Delta =\{\alpha _{1}\ldots \alpha _{n}\}}$ is a set of simple roots of ${\displaystyle \Phi }$.

6.  Cartan subgroup

7.  Cartan decomposition

Casimir

Casimir invariant, a distinguished element of a universal enveloping algebra.

Clebsch–Gordan coefficients

Clebsch–Gordan coefficients

center

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

center

1.  The center of a Lie group is the center of the group.

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

central series

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

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

Chevalley

1.  Claude Chevalley (1909 – 1984), a French mathematician

2.  A Chevalley basis is 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.

complex reflection group

complex reflection group

coroot

coroot

Coxeter

1.  H. S. M. Coxeter (1907 – 2003), a British-born Canadian geometer

2.  Coxeter group

3.  Coxeter number

derived algebra

1.  The derived algebra of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is ${\displaystyle [{\mathfrak {g}},{\mathfrak {g}}]}$. It is a subalgebra (in fact an ideal).

2.  A derived series is a sequence of ideals of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ obtained by repeatedly taking derived algebras; i.e., ${\displaystyle D^{0}{\mathfrak {g}}={\mathfrak {g}},D^{n}{\mathfrak {g}}=D^{n-1}{\mathfrak {g}}}$.

Dynkin

1.  Eugene Borisovich Dynkin (1924 – 2014), a Soviet and American mathematician

2.  

Dynkin diagrams.

extension

An exact sequence ${\displaystyle 0\to {\mathfrak {g}}'\to {\mathfrak {g}}\to {\mathfrak {g}}''\to 0}$ or ${\displaystyle {\mathfrak {g}}}$ is called a Lie algebra extension of ${\displaystyle {\mathfrak {g}}''}$ by ${\displaystyle {\mathfrak {g}}'}$.

exponential map

The exponential map for a Lie group G with ${\displaystyle {\mathfrak {g}}}$ is a map ${\displaystyle {\mathfrak {g}}\to G}$ which is not necessarily a homomorphism but satisfies a certain universal property.

exponential

E6, E7, E7½, E8, En, Exceptional Lie algebra

free Lie algebra

F

F4

fundamental

For "fundamental Weyl chamber", see #Weyl.

G

G2

generalized

1.  For "Generalized Cartan matrix", see #Cartan.

2.  For "Generalized Kac–Moody algebra", see #Kac–Moody algebra.

3.  For "Generalized Verma module", see #Verma.

group

Group analysis of differential equations.

homomorphism

1.  A Lie group homomorphism is a group homomorphism that is also a smooth map.

2.  A Lie algebra homomorphism is a linear map ${\displaystyle \phi :{\mathfrak {g}}_{1}\to {\mathfrak {g}}_{2}}$ such that ${\displaystyle \phi ([x,y])=[\phi (x),\phi (y)]\,\forall x,y\in {\mathfrak {g}}_{1}.}$

Harish-Chandra

1.  Harish-Chandra, (1923 – 1983), an Indian American mathematician and physicist

2.  Harish-Chandra homomorphism

3.  Harish-Chandra isomorphism

highest

1.  The theorem of the highest weight, stating the highest weights classify the irreducible representations.

2.  highest weight

3.  highest weight module

ideal

An ideal of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is a subspace ${\displaystyle {\mathfrak {g'}}}$ such that ${\displaystyle [{\mathfrak {g'}},{\mathfrak {g}}]\subseteq {\mathfrak {g'}}.}$ Unlike in ring theory, there is no distinguishability of left ideal and right ideal.

index

Index of a Lie algebra

invariant convex cone

An invariant convex cone is a closed convex cone in the Lie algebra of a connected Lie group that is invariant under inner automorphisms.

Iwasawa decomposition

Iwasawa decomposition

Jacobi identity

1.  

Carl Gustav Jacob Jacobi (1804 – 1851), a German mathematician.

2.  Given a binary operation ${\displaystyle [\cdot ,\,\cdot ]:V^{2}\to V}$, the Jacobi identity states: [[x, y], z] + [[y, z], x] + [[z, x], y] = 0.

Kac–Moody algebra

Kac–Moody algebra

Killing

1.  Wilhelm Killing (1847 – 1923), a German mathematician.

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

Kirillov

Kirillov character formula

Langlands

Langlands decomposition

Langlands dual

Lie

1.  

Sophus Lie (1842 – 1899), a Norwegian mathematician

2.  A Lie group is a group that has a compatible structure of a smooth manifold.

3.  A Lie algebra is a vector space ${\displaystyle {\mathfrak {g}}}$ over a field ${\displaystyle F}$ with a binary operation [·, ·] (called the Lie bracket or abbr. bracket) , which satisfies the following conditions: ${\displaystyle \forall a,b\in F,x,y,z\in {\mathfrak {g}}}$,

1. ${\displaystyle [ax+by,z]=a[x,z]+b[y,z]}$ (bilinearity)
2. ${\displaystyle [x,x]=0}$ (alternating)
3. ${\displaystyle [[x,y],z]+[[y,z],x]+[[z,x],y]=0}$ (Jacobi identity)

nilpotent

1.  A nilpotent Lie group.

2.  A nilpotent Lie algebra is a Lie algebra that is nilpotent as an ideal; i.e., some power is zero: ${\displaystyle [{\mathfrak {g}},[{\mathfrak {g}},[{\mathfrak {g}},\dots ,[{\mathfrak {g}},{\mathfrak {g}}]\dots ]]]=0}$.

3.  A nilpotent element of a semisimple Lie algebra^[1] is an element x such that the adjoint endomorphism ${\displaystyle ad_{x}}$ is a nilpotent endomorphism.

4.  A nilpotent cone

normalizer

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

maximal

1.  For "maximal compact subgroup", see #compact.

2.  For "maximal torus", see #torus.

parabolic

1.  Parabolic subgroup

2.  Parabolic subalgebra.

positive

For "positive root", see #positive.

quantum

quantum group.

quantized

quantized enveloping algebra.

radical

1.  The radical of a Lie group.

2.  The radical of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is the largest (i.e., unique maximal) solvable ideal of ${\displaystyle {\mathfrak {g}}}$.

real

real form.

reductive

1.  A reductive group.

2.  A reductive Lie algebra.

reflection

A reflection group, a group generated by reflections.

regular

1.  A regular element of a Lie algebra.

2.  A regular element with respect to a root system.

Let ${\displaystyle \Phi }$ be a root system. ${\displaystyle \gamma \in E}$ is called regular if ${\displaystyle (\gamma ,\alpha )\neq 0\,\forall \gamma \in \Phi }$.

For each set of simple roots ${\displaystyle \Delta }$ of ${\displaystyle \Phi }$, there exists a regular element ${\displaystyle \gamma \in E}$ such that ${\displaystyle (\gamma ,\alpha )>0\,\forall \gamma \in \Delta }$, conversely for each regular ${\displaystyle \gamma }$ there exist a unique set of base roots ${\displaystyle \Delta (\gamma )}$ such that the previous condition holds for ${\displaystyle \Delta =\Delta (\gamma )}$. It can be determined in following way: let ${\displaystyle \Phi ^{+}(\gamma )=\{\alpha \in \Phi |(\alpha ,\gamma )>0\}}$. Call an element ${\displaystyle \alpha }$ of ${\displaystyle \Phi ^{+}(\gamma )}$ decomposable if ${\displaystyle \alpha =\alpha '+\alpha ''}$ where ${\displaystyle \alpha ',\alpha ''\in \Phi ^{+}(\gamma )}$, then ${\displaystyle \Delta (\gamma )}$ is the set of all indecomposable elements of ${\displaystyle \Phi ^{+}(\gamma )}$

4.  Positive root of root system ${\displaystyle \Phi }$ with respect to a set of simple roots ${\displaystyle \Delta }$ is a root of ${\displaystyle \Phi }$ which is a linear combination of elements of ${\displaystyle \Delta }$ with nonnegative coefficients.

5.  Negative root of root system ${\displaystyle \Phi }$ with respect to a set of simple roots ${\displaystyle \Delta }$ is a root of ${\displaystyle \Phi }$ which is a linear combination of elements of ${\displaystyle \Delta }$ with nonpositive coefficients.

8.  inverse of a root system: Given a root system ${\displaystyle \Phi }$. Define ${\displaystyle \alpha ^{v}={\frac {2\alpha }{(\alpha ,\alpha )}}}$, ${\displaystyle \Phi ^{v}=\{\alpha ^{v}|\alpha \in \Phi \}}$ is called the inverse of a root system.

${\displaystyle \Phi ^{v}}$ is again a root system and have the identical Weyl group as ${\displaystyle \Phi }$.

Serre

Serre's theorem states that, given a (finite reduced) root system ${\displaystyle \Phi }$, there exists a unique (up to a choice of a base) semisimple Lie algebra whose root system is ${\displaystyle \Phi }$.

simple

1.  A simple Lie group is a connected Lie group that is not abelian which does not have nontrivial connected normal subgroups.

2.  A simple Lie algebra is a Lie algebra that is non abelian and has only two ideals, itself and ${\displaystyle \{0\}}$.

3.  simply laced group (a simple Lie group is simply laced when its Dynkin diagram is without multiple edges).

4.  simple root. A subset ${\displaystyle \Delta }$ of a root system ${\displaystyle \Phi }$ is called a set of simple roots if it satisfies the following conditions:

• ${\displaystyle \Delta }$ is a linear basis of ${\displaystyle E}$.
• Each element of ${\displaystyle \Phi }$ is a linear combination of elements of ${\displaystyle \Delta }$ with coefficients that are either all nonnegative or all nonpositive.

Classical Lie algebras:

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

Exceptional Lie algebras:

2.  A solvable Lie algebra is a Lie algebra ${\displaystyle {\mathfrak {g}}}$ such that ${\displaystyle D^{n}{\mathfrak {g}}=0}$ for some ${\displaystyle n\geq 0}$; where ${\displaystyle D{\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}]}$ denotes the derived algebra of ${\displaystyle {\mathfrak {g}}}$.

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

Tits

Tits cone.

toral

1.  toral Lie algebra

2.  maximal toral subalgebra

• Unitarian trick

• Verma module

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

• Bourbaki, N. (1981), Groupes et Algèbres de Lie, Éléments de Mathématique, Hermann
• 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
• Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
• Kac, Victor (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.
• Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.
• Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4.
• J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French)