View source for Glossary of Lie groups and Lie algebras

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This is a glossary for the terminology applied in the mathematical theories of Lie groups and [[Lie algebra]]s. 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|quantum group]].


{{Lie groups}}'''Notations''':
*Throughout the glossary, <math>( \cdot, \cdot )</math> denotes the inner product of a Euclidean space ''E'' and <math>\langle \cdot, \cdot \rangle</math> denotes the rescaled inner product
:: <math>\langle \beta, \alpha \rangle = \frac{(\beta, \alpha)}{(\alpha, \alpha)} \, \forall \alpha, \beta \in E. </math>

==A==
{{glossary}}
{{term|1=abelian}}
{{defn|no=1|1=An [[Abelian Lie group|abelian Lie group]] is a Lie group that is an abelian group.}}
{{defn|no=2|1=An [[Abelian Lie algebra|abelian Lie algebra]] is a Lie algebra such that <math>[x, y] = 0</math> for every <math>x, y</math> in the algebra.}}

{{term|1=adjoint}}
{{defn|no=1|1=An adjoint representation of a Lie group:
::<math>\operatorname{Ad} : G \to \operatorname{GL}(\mathfrak g)</math>
:such that <math>\operatorname{Ad}(g)</math> is the differential at the identity element of the conjugation <math>c_g : G \to G, x \mapsto g x g^{-1}</math>.}}
{{defn|no=2|1=An adjoint representation of a Lie algebra is a Lie algebra representation
:<math>\textrm{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})</math> where <math>\textrm{ad}(x)y = [x, y]</math>.}}

{{term|1=Ado}}
{{defn|1=[[Ado's theorem]]: Any finite-dimensional Lie algebra is isomorphic to a subalgebra of <math>\mathfrak{gl}_V</math> for some finite-dimensional vector space V.}}

{{term|1=affine}}
{{defn|no=1|An [[Affine Lie algebra|affine Lie algebra]] is a particular type of Kac–Moody algebra.}}
{{defn|no=2|An affine Weyl group.}}

{{term|1=analytic}}
{{defn|no=1|An analytic subgroup}}

{{term|1=automorphism}}
{{defn|no=1|An [[Automorphism of a Lie algebra|automorphism of a Lie algebra]] is a linear automorphism preserving the bracket.}}

{{glossary end}}

==B==
{{glossary}}
{{term|1=B}}
{{defn|no=1|1=[[(B, N) pair|(B, N) pair]]}}

{{term|1=Borel}}
{{defn|no=1|1=[[Biography:Armand Borel|Armand Borel]] (1923 – 2003), a Swiss mathematician}}
{{defn|no=2|1=A [[Borel subgroup]].}}
{{defn|no=3|1=A [[Borel subalgebra]] is a maximal solvable subalgebra.}}
{{defn|no=4|1=Borel-Bott-Weil theorem}}

{{term|1=Bruhat}}
{{defn|no=1|[[Bruhat decomposition]]}}

{{glossary end}}

==C==
{{glossary}}

{{term|1=Cartan}}
{{defn|no=1|[[Biography:Élie Cartan|Élie Cartan]] (1869 – 1951), a French mathematician}}
{{defn|no=2|A [[Cartan subalgebra]] <math>\mathfrak{h}</math> of a Lie algebra <math>\mathfrak{g}</math> is a nilpotent subalgebra satisfying <math>N_\mathfrak{g}(\mathfrak{h}) = \mathfrak{h}</math>.}}
{{defn|no=3|Cartan criterion for solvability: A Lie algebra <math>\mathfrak{g}</math> is solvable iff <math>\kappa( \mathfrak{g}, [\mathfrak{g},\mathfrak{g}] ) = 0</math>.}}
{{defn|no=4|[[Cartan's criterion#Cartan's criterion for semisimplicity|Cartan criterion for semisimplicity]]: (1) If <math>\kappa( \cdot, \cdot)</math> is nondegenerate, then <math>\mathfrak{g}</math> is semisimple. (2) If <math>\mathfrak{g}</math> is semisimple and the underlying field <math>F</math> has characteristic 0 , then <math>\kappa( \cdot, \cdot)</math> is nondegenerate.}}
{{defn|no=5|The [[Cartan matrix]] of the root system <math>\Phi</math> is the matrix <math>( \langle \alpha_i , \alpha_j \rangle )_{i,j=1}^n</math>, where <math>\Delta = \{\alpha_1 \ldots \alpha_n\}</math> is a set of simple roots of <math>\Phi</math>.}}
{{defn|no=6|[[Cartan subgroup]]}}
{{defn|no=7|[[Cartan decomposition]]}}

{{term|1=Casimir}}
{{defn|1=Casimir invariant, a distinguished element of a universal enveloping algebra.}}

{{term|1=Clebsch–Gordan coefficients}}
{{defn|1=[[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]]}}

{{term|1=center}}
{{defn|no=2|The centralizer of a subset <math>X</math> of a Lie algebra <math>\mathfrak{g}</math> is <math>C_{\mathfrak{g}}(X) := \{x \in \mathfrak{g} | [x, X] = \{0\} \}</math>.}}

{{term|1=center}}
{{defn|no=1|1=The center of a Lie group is the center of the group.}}
{{defn|no=2|1=The center of a Lie algebra is the centralizer of itself : <math>Z(L) := \{x \in \mathfrak{g} | [x, \mathfrak{g}] = 0 \}</math>}}

{{term|1=central series}}
{{defn|no=1|A descending central series (or lower central series) is a sequence of ideals of a Lie algebra <math>L</math> defined by <math>C^0(L) = L , \, C^1(L) = [L,L] , \, C^{n+1}(L) = [L, C^n(L)]</math>}}
{{defn|no=2|An ascending central series (or upper central series) is a sequence of ideals of a Lie algebra <math>L</math> defined by <math>C_0(L) = \{0\} , \, C_1(L) = Z(L)</math> (center of L) , <math>C_{n+1}(L) = \pi_n^{-1} ( Z ( L / C_{n}(L) ) )</math>, where <math>\pi_i</math> is the natural homomorphism <math>L \to L/C_n(L)</math>}}

{{term|1=Chevalley}}
{{defn|no=1|[[Biography:Claude Chevalley|Claude Chevalley]] (1909 – 1984), a French mathematician}}
{{defn|no=2|A [[Chevalley basis]] is a [[Basis (linear algebra)|basis]] constructed by [[Biography:Claude Chevalley|Claude Chevalley]] with the property that all [[Algebra over a field#Structure coefficients|structure constants]] are integers. Chevalley used these bases to construct analogues of [[Lie group]]s over [[Finite field|finite field]]s, called Chevalley groups.}}

{{term|1=complex reflection group}}
{{defn|1=[[Complex reflection group|complex reflection group]]}}

{{term|1=coroot}}
{{defn|1=coroot}}

{{term|1=Coxeter}}
{{defn|no=1|1=[[Biography:Harold Scott MacDonald Coxeter|H. S. M. Coxeter]] (1907 – 2003), a British-born Canadian geometer}}
{{defn|no=2|1=[[Coxeter group]]}}
{{defn|no=3|1=Coxeter number}}

{{glossary end}}

==D==
{{glossary}}
{{term|1=derived algebra}}
{{defn|no=1|1=The derived algebra of a Lie algebra <math>\mathfrak{g}</math> is <math>[\mathfrak{g}, \mathfrak{g} ]</math>. It is a subalgebra (in fact an ideal).}}
{{defn|no=2|1=A derived series is a sequence of ideals of a Lie algebra <math>\mathfrak g</math> obtained by repeatedly taking derived algebras; i.e., <math>D^0 \mathfrak{g} = \mathfrak{g}, D^n \mathfrak{g} = D^{n-1}\mathfrak{g}</math>.}}

{{term|1=Dynkin}}
{{defn|no=1|Eugene Borisovich Dynkin (1924 – 2014), a Soviet and American mathematician}}
{{defn|no=2|[[File:Finite Dynkin diagrams.svg|thumb|Dynkin diagrams]]Dynkin diagrams.}}

{{glossary end}}

==E==
{{glossary}}
{{term|1=extension}}
{{defn|1=An exact sequence <math>0 \to \mathfrak{g}' \to \mathfrak{g} \to \mathfrak{g}'' \to 0</math> or <math>\mathfrak{g}</math> is called a [[Physics:Lie algebra extension|Lie algebra extension]] of <math>\mathfrak{g}''</math> by <math>\mathfrak{g}'</math>.}}

{{term|1=exponential map}}
{{defn|1=The exponential map for a Lie group ''G'' with <math>\mathfrak g</math> is a map <math>\mathfrak g \to G</math> which is not necessarily a homomorphism but satisfies a certain universal property.}}

{{term|1=exponential}}
{{defn|1=[[E6 (mathematics)|E6]], [[E7 (mathematics)|E7]], E7½, [[E8 (mathematics)|E8]], [[En (Lie algebra)|En]], [[Exceptional Lie algebra]]}}

{{glossary end}}

==F==
{{glossary}}
{{term|1=free Lie algebra}}

{{term|1=F}}
{{defn|1=[[F4 (mathematics)|F4]]}}

{{term|fundamental}}
{{defn|1=For "fundamental Weyl chamber", see #Weyl.}}

{{glossary end}}

==G==
{{glossary}}

{{term|1=G}}
{{defn|1=[[G2 (mathematics)|G2]]}}

{{term|1=generalized}}
{{defn|no=1|1=For "Generalized Cartan matrix", see #Cartan.}}
{{defn|no=2|For "[[Generalized Kac–Moody algebra]]", see #Kac–Moody algebra.}}
{{defn|no=3|For "[[Generalized Verma module]]", see #Verma.}}

{{term|group}}
{{defn|1=[[Group analysis of differential equations]].}}

{{glossary end}}

==H==
{{glossary}}
{{term|1=homomorphism}}
{{defn|no=1|1=A Lie group homomorphism is a group homomorphism that is also a smooth map.}}
{{defn|no=2|1=A Lie algebra homomorphism is a linear map <math>\phi : \mathfrak{g}_1 \to \mathfrak{g}_2</math> such that <math>\phi([x,y]) = [ \phi(x), \phi(y) ] \, \forall x,y \in \mathfrak{g}_1.</math>}}

{{term|1=Harish-Chandra}}
{{defn|no=1|Harish-Chandra, (1923 – 1983), an Indian American mathematician and physicist}}
{{defn|no=2|[[Harish-Chandra homomorphism]]}}
{{defn|no=3|[[Harish-Chandra isomorphism]]}}

{{term|1=highest}}
{{defn|no=1|The [[Theorem of the highest weight|theorem of the highest weight]], stating the highest weights classify the irreducible representations.}}
{{defn|no=2|highest weight}}
{{defn|no=3|highest weight module}}

{{glossary end}}

==I==
{{glossary}}
{{term|1=ideal}}
{{defn|1=An ideal of a Lie algebra <math>\mathfrak{g}</math> is a subspace <math>\mathfrak{g'}</math> such that <math>[\mathfrak{g'}, \mathfrak{g}] \subseteq \mathfrak{g'}.</math> Unlike in ring theory, there is no distinguishability of left ideal and right ideal.}}
{{term|1=index}}
{{defn|1=[[Index of a Lie algebra]]}}

{{term|1=invariant convex cone}}
{{defn|1=An [[Invariant convex cone|invariant convex cone]] is a closed convex cone in the Lie algebra of a connected Lie group that is invariant under inner automorphisms.}}

{{term|1=Iwasawa decomposition}}
{{defn|1=[[Iwasawa decomposition]]}}

{{glossary end}}

== J ==
{{glossary}}
{{term|1=Jacobi identity}}
{{defn|no=1|1=[[File:Carl Jacobi.jpg|thumb|Carl Gustav Jacob Jacobi]][[Biography:Carl Gustav Jacob Jacobi|Carl Gustav Jacob Jacobi]] (1804 – 1851), a German mathematician.}}
{{defn|no=2|1=Given a binary operation <math>[\cdot,\,\cdot ] : V^2 \to V</math>, the [[Jacobi identity]] states: [<nowiki />[''x'', ''y''], ''z''] + [<nowiki />[''y'', ''z''], ''x''] + [<nowiki />[''z'', ''x''], ''y''] = 0.}}

{{glossary end}}

== K ==
{{glossary}}

{{term|1=Kac–Moody algebra}}
{{defn|1=[[Kac–Moody algebra]]}}

{{term|1=Killing}}
{{defn|no=1|Wilhelm Killing (1847 – 1923), a German mathematician.}}
{{defn|no=2|The [[Killing form]] on a Lie algebra <math>\mathfrak{g}</math> is a symmetric, associative, bilinear form defined by <math>\kappa(x, y) := \textrm{Tr}( \textrm{ad}\,x\, \textrm{ad}\, y )\  \forall x,y \in \mathfrak{g}</math>.}}

{{term|1=Kirillov}}
{{defn|1=[[Kirillov character formula]]}}

{{glossary end}}

==L==
{{glossary}}
{{term|1=Langlands}}
{{defn|1=[[Langlands decomposition]]}}
{{defn|1=Langlands dual}}

{{term|1=Lie}}
{{defn|no=1|1=[[File:Portrett av Sophus Lie.jpg|thumb|Sophus Lie]][[Biography:Sophus Lie|Sophus Lie]] (1842 – 1899), a Norwegian mathematician}}
{{defn|no=2|1=A [[Lie group]] is a group that has a compatible structure of a smooth manifold.}}
{{defn|no=3|1=A [[Lie algebra]] is a vector space <math>\mathfrak{g}</math> over a field <math>F</math> with a binary operation  [·, ·] (called the '''Lie bracket''' or abbr. '''bracket''') , which satisfies the following conditions: <math>\forall a,b \in F, x,y,z \in \mathfrak{g}</math>,
:# <math>[ax+by,z] = a[x,z] + b[y,z]</math> (bilinearity)
:# <math>[x,x] = 0</math> (alternating)
:# <math>x,y], z ] + [[y,z],x] + [[z,x],y] = 0</math> ([[Jacobi identity)}}
{{defn|no=4|1=[[Lie group–Lie algebra correspondence]]}}
{{defn|no=5|1=[[Lie's theorem]]
: Let <math>\mathfrak{g}</math> be a finite-dimensional complex [[Solvable Lie algebra|solvable Lie algebra]] over [[Algebraically closed field|algebraically closed field]] of characteristic <math>0</math>, and let <math>V</math> be a nonzero finite dimensional representation of  <math>\mathfrak{g}</math>.  Then there exists an element of <math>V</math> which is a simultaneous eigenvector for all elements of  <math>\mathfrak{g}</math>.}}
{{defn|no=6|1=Compact Lie group.}}
{{defn|no=7|1=Semisimple Lie group; see #semisimple.}}

{{term|1=Levi}}
{{defn|1=[[Levi decomposition]]}}

{{glossary end}}

==N==
{{glossary}}
{{term|1=nilpotent}}
{{defn|no=1|1=A nilpotent Lie group.}}
{{defn|no=2|1=A [[Nilpotent Lie algebra|nilpotent Lie algebra]] is a Lie algebra that is [[Nilpotent ideal|nilpotent]] as an ideal; i.e., some power is zero: <math>[\mathfrak g, [\mathfrak g, [\mathfrak g, \dots, [\mathfrak g, \mathfrak g] \dots ]]] = 0</math>.}}
{{defn|no=3|1=A nilpotent element of a semisimple Lie algebra<ref>Editorial note: the definition of a nilpotent element in a general Lie algebra seems unclear.</ref> is an element ''x'' such that the adjoint endomorphism <math>ad_x</math> is a nilpotent endomorphism.}}
{{defn|no=4|A [[Nilpotent cone|nilpotent cone]]}}

{{term|1=normalizer}}
{{defn|1=The normalizer of a subspace <math>K</math> of a Lie algebra <math>\mathfrak{g}</math> is <math>N_{\mathfrak{g}}(K) := \{x \in \mathfrak{g} | [x, K] \subseteq K \}</math>.}}

{{glossary end}}

==M==
{{glossary}}

{{term|1=maximal}}
{{defn|no=1|For "[[Maximal compact subgroup|maximal compact subgroup]]", see #compact.}}
{{defn|no=2|For "[[Maximal torus|maximal torus]]", see #torus.}}

{{glossary end}}

==P==
{{glossary}}

{{term|1=parabolic}}
{{defn|no=1|1=[[Borel subgroup|Parabolic subgroup]]}}
{{defn|no=2|1=Parabolic subalgebra.}}

{{term|1=positive}}
{{defn|1=For "positive root", see #positive.}}

{{glossary end}}

==Q==

{{glossary}}

{{term|1=quantum}}
{{defn|1=[[Quantum group|quantum group]].}}

{{term|1=quantized}}
{{defn|1=[[Quantized enveloping algebra|quantized enveloping algebra]].}}

{{glossary end}}

==R==
{{glossary}}
{{term|1=radical}}
{{defn|no=1|1=The radical of a Lie group.}}
{{defn|no=2|1=The [[Radical of a Lie algebra|radical of a Lie algebra]] <math>\mathfrak g</math> is the largest (i.e., unique maximal) solvable ideal of <math>\mathfrak{g}</math>.}}

{{term|1=real}}
{{defn|1=real form.}}

{{term|1=reductive}}
{{defn|no=1|1=A [[Reductive group|reductive group]].}}
{{defn|no=2|1=A [[Reductive Lie algebra|reductive Lie algebra]].}}

{{term|1=reflection}}
{{defn|1=A [[Reflection group|reflection group]], a group generated by reflections.}}

{{term|1=regular}}
{{defn|no=1|1=A [[Regular element of a Lie algebra|regular element of a Lie algebra]].}}
{{defn|no=2|1=A regular element with respect to a root system.
: Let <math>\Phi</math> be a root system. <math>\gamma \in E</math> is called regular if <math>(\gamma, \alpha) \ne 0 \, \forall \gamma \in \Phi</math>.
: For each set of simple roots <math>\Delta</math> of <math>\Phi</math>, there exists a regular element <math>\gamma \in E</math> such that <math> ( \gamma, \alpha ) > 0 \, \forall \gamma \in \Delta</math>, conversely for each regular <math>\gamma</math> there exist a unique set of base roots <math>\Delta(\gamma)</math> such that the previous condition holds for <math>\Delta = \Delta(\gamma)</math>. It can be determined in following way: let <math>\Phi^+(\gamma) = \{\alpha \in \Phi | (\alpha, \gamma) > 0\}</math>. Call an element <math>\alpha</math> of <math>\Phi^+(\gamma)</math> decomposable if <math>\alpha = \alpha' + \alpha''</math> where <math>\alpha', \alpha'' \in \Phi^+(\gamma)</math>, then <math>\Delta(\gamma)</math>  is the set of all indecomposable elements of <math>\Phi^+(\gamma)</math>}}

{{term|1=root}}
{{defn|no=1|1=root of a semisimple Lie algebra:
: Let <math>\mathfrak{g}</math> be a semisimple Lie algebra, <math>\mathfrak{h}</math> be a Cartan subalgebra of <math>\mathfrak{g}</math>. For <math>\alpha \in \mathfrak{h}^*</math>, let <math>\mathfrak{g_\alpha} := \{ x \in \mathfrak{g} | [h,x] = \alpha(h) x \, \forall h \in \mathfrak{h} \} </math>. <math>\alpha</math> is called a root of <math>\mathfrak{g}</math> if it is nonzero and <math>\mathfrak{g_\alpha} \ne \{0\} </math>
: The set of all roots is denoted by <math>\Phi</math> ; it forms a root system.}}
{{defn|no=2|1=[[Root system]]
: A subset <math>\Phi</math> of the Euclidean space <math>E</math> is called a root system if it satisfies the following conditions:
:* <math>\Phi</math> is finite, <math>\textrm{span} (\Phi) = E</math> and <math>0 \notin \Phi</math>.
:* For all <math>\alpha \in \Phi</math> and <math>c \in \mathbb{R}</math>, <math>c \alpha \in \Phi</math> iff <math>c = \pm 1</math>.
:* For all <math>\alpha,\beta \in \Phi</math>, <math>\langle \alpha, \beta \rangle</math> is an integer.
:* For all <math>\alpha,\beta \in \Phi</math>, <math>S_\alpha(\beta)\in \Phi</math>, where <math>S_\alpha</math> is the reflection through the hyperplane normal to <math>\alpha</math>, i.e. <math>S_\alpha(x) = x - \langle x , \alpha \rangle \alpha</math>.}}
{{defn|no=3|[[Root datum]]}}
{{defn|no=4|Positive root of root system <math>\Phi</math> with respect to a set of simple roots <math>\Delta</math> is a root of <math>\Phi</math> which is a linear combination of elements of <math>\Delta</math> with nonnegative coefficients.}}
{{defn|no=5|Negative root of root system <math>\Phi</math> with respect to a set of simple roots <math>\Delta</math> is a root of <math>\Phi</math> which is a linear combination of elements of <math>\Delta</math> with nonpositive coefficients.}}
{{defn|no=6|long root}}
{{defn|no=7|short root}}
{{defn|no=8|inverse of a root system: Given a root system <math>\Phi</math>. Define <math>\alpha^v = \frac{2 \alpha}{ ( \alpha, \alpha) }</math>, <math>\Phi^v = \{ \alpha^v | \alpha \in \Phi \} </math> is called the inverse of a root system.
: <math>\Phi^v</math> is again a root system and have the identical Weyl group as <math> \Phi </math>.}}
{{defn|no=9|base of a root system: synonymous to "set of simple roots"}}
{{defn|no=10|dual of a root system: synonymous to "inverse of a root system"}}

{{glossary end}}

==S==
{{glossary}}
{{term|1=Serre}}
{{defn|1=[[Serre's theorem on a semisimple Lie algebra|Serre's theorem]] states that, given a (finite reduced) root system <math>\Phi</math>, there exists a unique (up to a choice of a base) semisimple Lie algebra whose root system is <math>\Phi</math>.}}

{{term|1=simple}}
{{defn|no=1|1=A [[Simple Lie group|simple Lie group]] is a connected Lie group that is not abelian which does not have nontrivial connected normal subgroups.}}
{{defn|no=2|1=A [[Simple Lie algebra|simple Lie algebra]] is a Lie algebra that is non abelian and has only two ideals, itself and <math>\{0\}</math>.}}
{{defn|no=3|simply laced group (a simple Lie group is simply laced when its Dynkin diagram is without multiple edges).}}
{{defn|no=4|simple root. A subset <math>\Delta</math> of a root system <math>\Phi</math> is called a set of simple roots if it satisfies the following conditions:
:* <math>\Delta</math> is a linear basis of <math>E</math>.
:* Each element of <math>\Phi</math> is a linear combination of elements of <math>\Delta</math> with coefficients that are either all nonnegative or all nonpositive.}}<!--; a partial order on the Eucliean space <math>E</math> defined by the set of simple root
: <math>\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 </math>-->
{{defn|no=5|content=Classification of simple Lie algebras}}

'''Classical Lie algebras''':

{| class="wikitable"
|-
| Special linear algebra
| <math>A_l \  (l \ge 1)</math>
| <math>l^2 + 2l</math>
| <math>\mathfrak{sl}(l+1, F) = \{ x \in \mathfrak{gl}(l+1,F) | Tr(x) = 0 \}</math> (traceless matrices)
|-
| Orthogonal algebra
| <math>B_l \  (l \ge 1)</math>
| <math>2 l^2 + l</math>
| <math>\mathfrak{o}(2l+1, F) = \{ x \in \mathfrak{gl}(2l+1,F) | s x = - x^t s , s = \begin{pmatrix}   1 & 0 & 0 \\ 0 & 0 & I_l \\ 0 & I_l & 0  \end{pmatrix}\}</math>
|-
| Symplectic algebra
| <math>C_l \  (l \ge 2)</math>
| <math>2 l^2 - l</math>
| <math>\mathfrak{sp}(2l, F) = \{ x \in \mathfrak{gl}(2l,F) | s x = - x^t s, s = \begin{pmatrix}   0 & I_l \\  -I_l & 0  \end{pmatrix}\}</math>
|-
| Orthogonal algebra
| <math>D_l (l \ge 1)</math>
| <math>2 l^2 + l</math>
| <math>\mathfrak{o}(2l, F) = \{ x \in \mathfrak{gl}(2l,F) | s x = - x^t s, s = \begin{pmatrix}   0 & I_l \\   I_l & 0  \end{pmatrix}\}</math>
|}

'''Exceptional Lie algebras''':

{| class="wikitable"
|-
! Root System
! dimension
|-
| [[G2 (mathematics)|G<sub>2</sub>]]
| 14
|-
| [[F4 (mathematics)|F<sub>4</sub>]]
| 52
|-
| [[E6 (mathematics)|E<sub>6</sub>]]
| 78
|-
| [[E7 (mathematics)|E<sub>7</sub>]]
| 133
|-
| [[E8 (mathematics)|E<sub>8</sub>]]
| 248
|}

{{term|1=semisimple}}
{{defn|no=1|1=A semisimple Lie group}}
{{defn|no=2|1=A [[Semisimple Lie algebra|semisimple Lie algebra]] is a nonzero Lie algebra that has no nonzero abelian ideal.}}
{{defn|no=3|1=In a [[Semisimple Lie algebra|semisimple Lie algebra]], an element is semisimple if its image under the adjoint representation is semisimple; see [[Semisimple Lie algebra]].}}

{{term|1=solvable}}
{{defn|no=1|1=A solvable Lie group}}
{{defn|no=2|1=A [[Solvable Lie algebra|solvable Lie algebra]] is a Lie algebra <math>\mathfrak g</math> such that <math>D^n \mathfrak g = 0</math> for some <math>n \ge 0</math>; where <math>D \mathfrak g = [\mathfrak g, \mathfrak g]</math> denotes the derived algebra of <math>\mathfrak g</math>.}}

{{term|1=split}}

{{term|1=Stiefel}}
{{defn|1=Stiefel diagram of a compact connected Lie group.}}

{{term|1=subalgebra}}
{{defn|1=A subspace <math>\mathfrak{g'}</math> of a Lie algebra <math>\mathfrak{g}</math> is called the subalgebra of <math>\mathfrak{g}</math> if it is closed under bracket, i.e. <math>[\mathfrak{g'}, \mathfrak{g'}] \subseteq \mathfrak{g'}.</math>}}

{{glossary end}}

==T==
{{glossary}}

{{term|1=Tits}}
{{defn|Tits cone.}}

{{term|1=toral}}
{{defn|no=1|1=[[Toral Lie algebra|toral Lie algebra]]}}
{{defn|no=2|1=maximal toral subalgebra}}

{{glossary end}}

==U==
{{glossary}}

* [[Unitarian trick]]

{{glossary end}}

==V==
{{glossary}}

* [[Verma module]]

{{glossary end}}

==W==
{{term|1=Weyl}}
{{defn|no=1|1=[[Biography:Hermann Weyl|Hermann Weyl]] (1885 – 1955), a German mathematician}}
{{defn|no=2|A Weyl chamber is one of the connected components of the complement in ''V'', a real vector space on which a root system is defined, when the hyperplanes orthogonal to the root vectors are removed.}}
{{defn|no=3|The [[Weyl character formula]] gives in closed form the characters of the irreducible complex representations of the simple Lie groups.}}
{{defn|no=4|[[Weyl group]]: Weyl group of a root system <math>\Phi</math> is a (necessarily finite) group of orthogonal linear transformations of <math>E</math> which is generated by reflections through hyperplanes normal to roots of <math>\Phi</math>}}

==References==
{{reflist}}
* {{citation|last=Bourbaki|first=N.|series=Éléments de Mathématique|title=Groupes et Algèbres de Lie|publisher=Hermann|year=1981}}
* [[Biography:Karin Erdmann|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}}
* [[Biography:Nathan Jacobson|Jacobson, Nathan]], ''Lie algebras'', Republication of the 1962 original. Dover Publications, Inc., New York, 1979.  {{isbn|0-486-63832-4}}
*{{cite book|first=Victor|last=Kac|title=Infinite dimensional Lie algebras|edition= 3rd |publisher= {{wipe|Cambridge University Press}} |year=1990|isbn=0-521-46693-8|url=https://books.google.com/books?id=kuEjSb9teJwC&q=Victor%20G.%20Kac&pg=PP1}}
* [[Biography:Claudio Procesi|Claudio Procesi]] (2007) ''Lie Groups: an approach through invariants and representation'', Springer, {{isbn|9780387260402}}.
*{{Citation |url=https://books.google.com/books?id=7AHsSUrooSsC&pg=PA3|title=Algèbres de Lie semi-simples complexes|last=Serre|first=Jean-Pierre|date=2000|publisher=Springer|trans-title=Complex Semisimple Lie Algebras|isbn=978-3-540-67827-4|language=en|ref={{Harvid|Serre}}|translator-last=Jones|translator-first=G. A.}}.
*J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French)

{{DEFAULTSORT:Glossary Of Lie Algebras}}
[[Category:Glossaries of mathematics|Lie Algebra]]
[[Category:Lie algebras| ]]

{{Sourceattribution|Glossary of Lie groups and Lie algebras}}