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 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.
| Group theory → Lie groups Lie groups |
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Notations:
- [math]\displaystyle{ \langle \beta, \alpha \rangle = \frac{(\beta, \alpha)}{(\alpha, \alpha)} \, \forall \alpha, \beta \in E. }[/math]
A
- 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 [math]\displaystyle{ [x, y] = 0 }[/math] for every [math]\displaystyle{ x, y }[/math] in the algebra.
- adjoint
- 1. An adjoint representation of a Lie group:
- [math]\displaystyle{ \operatorname{Ad} : G \to \operatorname{GL}(\mathfrak g) }[/math]
- such that [math]\displaystyle{ \operatorname{Ad}(g) }[/math] is the differential at the identity element of the conjugation [math]\displaystyle{ c_g : G \to G, x \mapsto g x g^{-1} }[/math].
- [math]\displaystyle{ \textrm{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) }[/math] where [math]\displaystyle{ \textrm{ad}(x)y = [x, y] }[/math].
B
- 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
C
- Cartan
- 1. Élie Cartan (1869 – 1951), a French mathematician
- 2. A Cartan subalgebra [math]\displaystyle{ \mathfrak{h} }[/math] of a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is a nilpotent subalgebra satisfying [math]\displaystyle{ N_\mathfrak{g}(\mathfrak{h}) = \mathfrak{h} }[/math].
- 3. Cartan criterion for solvability: A Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is solvable iff [math]\displaystyle{ \kappa( \mathfrak{g}, [\mathfrak{g},\mathfrak{g}] ) = 0 }[/math].
- 4. Cartan criterion for semisimplicity: (1) If [math]\displaystyle{ \kappa( \cdot, \cdot) }[/math] is nondegenerate, then [math]\displaystyle{ \mathfrak{g} }[/math] is semisimple. (2) If [math]\displaystyle{ \mathfrak{g} }[/math] is semisimple and the underlying field [math]\displaystyle{ F }[/math] has characteristic 0 , then [math]\displaystyle{ \kappa( \cdot, \cdot) }[/math] is nondegenerate.
- 5. The Cartan matrix of the root system [math]\displaystyle{ \Phi }[/math] is the matrix [math]\displaystyle{ ( \langle \alpha_i , \alpha_j \rangle )_{i,j=1}^n }[/math], where [math]\displaystyle{ \Delta = \{\alpha_1 \ldots \alpha_n\} }[/math] is a set of simple roots of [math]\displaystyle{ \Phi }[/math].
- 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 [math]\displaystyle{ X }[/math] of a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is [math]\displaystyle{ C_{\mathfrak{g}}(X) := \{x \in \mathfrak{g} | [x, X] = \{0\} \} }[/math].
- 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 : [math]\displaystyle{ Z(L) := \{x \in \mathfrak{g} | [x, \mathfrak{g}] = 0 \} }[/math]
- central series
- 1. A descending central series (or lower central series) is a sequence of ideals of a Lie algebra [math]\displaystyle{ L }[/math] defined by [math]\displaystyle{ C^0(L) = L , \, C^1(L) = [L,L] , \, C^{n+1}(L) = [L, C^n(L)] }[/math]
- 2. An ascending central series (or upper central series) is a sequence of ideals of a Lie algebra [math]\displaystyle{ L }[/math] defined by [math]\displaystyle{ C_0(L) = \{0\} , \, C_1(L) = Z(L) }[/math] (center of L) , [math]\displaystyle{ C_{n+1}(L) = \pi_n^{-1} ( Z ( L / C_{n}(L) ) ) }[/math], where [math]\displaystyle{ \pi_i }[/math] is the natural homomorphism [math]\displaystyle{ L \to L/C_n(L) }[/math]
- 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
D
- derived algebra
- 1. The derived algebra of a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is [math]\displaystyle{ [\mathfrak{g}, \mathfrak{g} ] }[/math]. It is a subalgebra (in fact an ideal).
- 2. A derived series is a sequence of ideals of a Lie algebra [math]\displaystyle{ \mathfrak g }[/math] obtained by repeatedly taking derived algebras; i.e., [math]\displaystyle{ D^0 \mathfrak{g} = \mathfrak{g}, D^n \mathfrak{g} = D^{n-1}\mathfrak{g} }[/math].
- Dynkin
- 1. Eugene Borisovich Dynkin (1924 – 2014), a Soviet and American mathematician
- 2. Dynkin diagrams.
E
- extension
- An exact sequence [math]\displaystyle{ 0 \to \mathfrak{g}' \to \mathfrak{g} \to \mathfrak{g}'' \to 0 }[/math] or [math]\displaystyle{ \mathfrak{g} }[/math] is called a Lie algebra extension of [math]\displaystyle{ \mathfrak{g}'' }[/math] by [math]\displaystyle{ \mathfrak{g}' }[/math].
- exponential map
- The exponential map for a Lie group G with [math]\displaystyle{ \mathfrak g }[/math] is a map [math]\displaystyle{ \mathfrak g \to G }[/math] which is not necessarily a homomorphism but satisfies a certain universal property.
- exponential
- E6, E7, E7½, E8, En, Exceptional Lie algebra
F
- free Lie algebra
- F
- F4
- fundamental
- For "fundamental Weyl chamber", see #Weyl.
G
- 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.
H
- 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 [math]\displaystyle{ \phi : \mathfrak{g}_1 \to \mathfrak{g}_2 }[/math] such that [math]\displaystyle{ \phi([x,y]) = [ \phi(x), \phi(y) ] \, \forall x,y \in \mathfrak{g}_1. }[/math]
- 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
I
- ideal
- An ideal of a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is a subspace [math]\displaystyle{ \mathfrak{g'} }[/math] such that [math]\displaystyle{ [\mathfrak{g'}, \mathfrak{g}] \subseteq \mathfrak{g'}. }[/math] 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
J
- Jacobi identity
- 1. Carl Gustav Jacob Jacobi (1804 – 1851), a German mathematician.
- 2. Given a binary operation [math]\displaystyle{ [\cdot,\,\cdot ] : V^2 \to V }[/math], the Jacobi identity states: [[x, y], z] + [[y, z], x] + [[z, x], y] = 0.
K
- Kac–Moody algebra
- Kac–Moody algebra
- Killing
- 1. Wilhelm Killing (1847 – 1923), a German mathematician.
- 2. The Killing form on a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is a symmetric, associative, bilinear form defined by [math]\displaystyle{ \kappa(x, y) := \textrm{Tr}( \textrm{ad}\,x\, \textrm{ad}\, y )\ \forall x,y \in \mathfrak{g} }[/math].
- Kirillov
- Kirillov character formula
L
- 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. {{defn|no=3|1=A Lie algebra is a vector space [math]\displaystyle{ \mathfrak{g} }[/math] over a field [math]\displaystyle{ F }[/math] with a binary operation [·, ·] (called the Lie bracket or abbr. bracket) , which satisfies the following conditions: [math]\displaystyle{ \forall a,b \in F, x,y,z \in \mathfrak{g} }[/math],
- [math]\displaystyle{ [ax+by,z] = a[x,z] + b[y,z] }[/math] (bilinearity)
- [math]\displaystyle{ [x,x] = 0 }[/math] (alternating)
- [math]\displaystyle{ x,y], z ] + [[y,z],x] + [[z,x],y] = 0 }[/math] ([[Jacobi identity)}}
- 4. Lie group–Lie algebra correspondence
- 5. Lie's theorem
- Let [math]\displaystyle{ \mathfrak{g} }[/math] be a finite-dimensional complex solvable Lie algebra over algebraically closed field of characteristic [math]\displaystyle{ 0 }[/math], and let [math]\displaystyle{ V }[/math] be a nonzero finite dimensional representation of [math]\displaystyle{ \mathfrak{g} }[/math]. Then there exists an element of [math]\displaystyle{ V }[/math] which is a simultaneous eigenvector for all elements of [math]\displaystyle{ \mathfrak{g} }[/math].
N
- 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: [math]\displaystyle{ [\mathfrak g, [\mathfrak g, [\mathfrak g, \dots, [\mathfrak g, \mathfrak g] \dots ]]] = 0 }[/math].
- 3. A nilpotent element of a semisimple Lie algebra[1] is an element x such that the adjoint endomorphism [math]\displaystyle{ ad_x }[/math] is a nilpotent endomorphism.
- 4. A nilpotent cone
- normalizer
- The normalizer of a subspace [math]\displaystyle{ K }[/math] of a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is [math]\displaystyle{ N_{\mathfrak{g}}(K) := \{x \in \mathfrak{g} | [x, K] \subseteq K \} }[/math].
M
- maximal
- 1. For "maximal compact subgroup", see #compact.
- 2. For "maximal torus", see #torus.
P
- parabolic
- 1. Parabolic subgroup
- 2. Parabolic subalgebra.
- positive
- For "positive root", see #positive.
Q
- quantum
- quantum group.
- quantized
- quantized enveloping algebra.
R
- radical
- 1. The radical of a Lie group.
- 2. The radical of a Lie algebra [math]\displaystyle{ \mathfrak g }[/math] is the largest (i.e., unique maximal) solvable ideal of [math]\displaystyle{ \mathfrak{g} }[/math].
- 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 [math]\displaystyle{ \Phi }[/math] be a root system. [math]\displaystyle{ \gamma \in E }[/math] is called regular if [math]\displaystyle{ (\gamma, \alpha) \ne 0 \, \forall \gamma \in \Phi }[/math].
- For each set of simple roots [math]\displaystyle{ \Delta }[/math] of [math]\displaystyle{ \Phi }[/math], there exists a regular element [math]\displaystyle{ \gamma \in E }[/math] such that [math]\displaystyle{ ( \gamma, \alpha ) \gt 0 \, \forall \gamma \in \Delta }[/math], conversely for each regular [math]\displaystyle{ \gamma }[/math] there exist a unique set of base roots [math]\displaystyle{ \Delta(\gamma) }[/math] such that the previous condition holds for [math]\displaystyle{ \Delta = \Delta(\gamma) }[/math]. It can be determined in following way: let [math]\displaystyle{ \Phi^+(\gamma) = \{\alpha \in \Phi | (\alpha, \gamma) \gt 0\} }[/math]. Call an element [math]\displaystyle{ \alpha }[/math] of [math]\displaystyle{ \Phi^+(\gamma) }[/math] decomposable if [math]\displaystyle{ \alpha = \alpha' + \alpha'' }[/math] where [math]\displaystyle{ \alpha', \alpha'' \in \Phi^+(\gamma) }[/math], then [math]\displaystyle{ \Delta(\gamma) }[/math] is the set of all indecomposable elements of [math]\displaystyle{ \Phi^+(\gamma) }[/math]
- Let [math]\displaystyle{ \mathfrak{g} }[/math] be a semisimple Lie algebra, [math]\displaystyle{ \mathfrak{h} }[/math] be a Cartan subalgebra of [math]\displaystyle{ \mathfrak{g} }[/math]. For [math]\displaystyle{ \alpha \in \mathfrak{h}^* }[/math], let [math]\displaystyle{ \mathfrak{g_\alpha} := \{ x \in \mathfrak{g} | [h,x] = \alpha(h) x \, \forall h \in \mathfrak{h} \} }[/math]. [math]\displaystyle{ \alpha }[/math] is called a root of [math]\displaystyle{ \mathfrak{g} }[/math] if it is nonzero and [math]\displaystyle{ \mathfrak{g_\alpha} \ne \{0\} }[/math]
- The set of all roots is denoted by [math]\displaystyle{ \Phi }[/math] ; it forms a root system.
- A subset [math]\displaystyle{ \Phi }[/math] of the Euclidean space [math]\displaystyle{ E }[/math] is called a root system if it satisfies the following conditions:
- [math]\displaystyle{ \Phi }[/math] is finite, [math]\displaystyle{ \textrm{span} (\Phi) = E }[/math] and [math]\displaystyle{ 0 \notin \Phi }[/math].
- For all [math]\displaystyle{ \alpha \in \Phi }[/math] and [math]\displaystyle{ c \in \mathbb{R} }[/math], [math]\displaystyle{ c \alpha \in \Phi }[/math] iff [math]\displaystyle{ c = \pm 1 }[/math].
- For all [math]\displaystyle{ \alpha,\beta \in \Phi }[/math], [math]\displaystyle{ \langle \alpha, \beta \rangle }[/math] is an integer.
- For all [math]\displaystyle{ \alpha,\beta \in \Phi }[/math], [math]\displaystyle{ S_\alpha(\beta)\in \Phi }[/math], where [math]\displaystyle{ S_\alpha }[/math] is the reflection through the hyperplane normal to [math]\displaystyle{ \alpha }[/math], i.e. [math]\displaystyle{ S_\alpha(x) = x - \langle x , \alpha \rangle \alpha }[/math].
- [math]\displaystyle{ \Phi^v }[/math] is again a root system and have the identical Weyl group as [math]\displaystyle{ \Phi }[/math].
S
- Serre
- Serre's theorem states that, given a (finite reduced) root system [math]\displaystyle{ \Phi }[/math], there exists a unique (up to a choice of a base) semisimple Lie algebra whose root system is [math]\displaystyle{ \Phi }[/math].
- 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 [math]\displaystyle{ \{0\} }[/math].
- 3. simply laced group (a simple Lie group is simply laced when its Dynkin diagram is without multiple edges).
- 4. simple root. A subset [math]\displaystyle{ \Delta }[/math] of a root system [math]\displaystyle{ \Phi }[/math] is called a set of simple roots if it satisfies the following conditions:
- [math]\displaystyle{ \Delta }[/math] is a linear basis of [math]\displaystyle{ E }[/math].
- Each element of [math]\displaystyle{ \Phi }[/math] is a linear combination of elements of [math]\displaystyle{ \Delta }[/math] with coefficients that are either all nonnegative or all nonpositive.
Classical Lie algebras:
| Special linear algebra | [math]\displaystyle{ A_l \ (l \ge 1) }[/math] | [math]\displaystyle{ l^2 + 2l }[/math] | [math]\displaystyle{ \mathfrak{sl}(l+1, F) = \{ x \in \mathfrak{gl}(l+1,F) | Tr(x) = 0 \} }[/math] (traceless matrices) |
| Orthogonal algebra | [math]\displaystyle{ B_l \ (l \ge 1) }[/math] | [math]\displaystyle{ 2 l^2 + l }[/math] | [math]\displaystyle{ \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]\displaystyle{ C_l \ (l \ge 2) }[/math] | [math]\displaystyle{ 2 l^2 - l }[/math] | [math]\displaystyle{ \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]\displaystyle{ D_l (l \ge 1) }[/math] | [math]\displaystyle{ 2 l^2 + l }[/math] | [math]\displaystyle{ \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:
| Root System | dimension |
|---|---|
| G2 | 14 |
| F4 | 52 |
| E6 | 78 |
| E7 | 133 |
| E8 | 248 |
T
- Tits
- Tits cone.
- toral
- 1. toral Lie algebra
- 2. maximal toral subalgebra
U
V
W
References
- ↑ Editorial note: the definition of a nilpotent element in a general Lie algebra seems unclear.
- 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. https://books.google.com/books?id=kuEjSb9teJwC&q=Victor%20G.%20Kac&pg=PP1.
- Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.
- Serre, Jean-Pierre (2000) (in en), Algèbres de Lie semi-simples complexes, Springer, ISBN 978-3-540-67827-4, https://books.google.com/books?id=7AHsSUrooSsC&pg=PA3.
- J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French)
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