...des with the rank of its Lie algebra (see [[Rank of a Lie algebra]]). If a Lie group $G$ coincides with the set of real or complex points of a [[linear al
...TD valign="top">[a1]</TD> <TD valign="top"> R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) Chapt. 1 {{ISBN|0-471-13735-9}}</TD></T
1 KB (181 words) - 10:43, 19 May 2024
''Lie group of type $(E)$''
...$\exp\colon \mathfrak{g} \to G$, where $\mathfrak{g}$ is the [[Lie algebra|Lie algebra]] of $G$, is a [[Diffeomorphism|diffeomorphism]].
2 KB (253 words) - 22:20, 14 November 2014
''triangular Lie group''
...n (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) are real for any element $g$.
2 KB (303 words) - 18:20, 12 December 2019
...Lie algebras have with groups, these concepts have been mainly studied for Lie algebras.
...and algebras]]; [[Non-associative rings and algebras|Non-associative rings and algebras]]) is the intersection of its maximal subalgebras. Unlike the grou
2 KB (323 words) - 13:52, 25 April 2014
$#C+1 = 13 : ~/encyclopedia/old_files/data/L058/L.0508600 Lie group, Banach
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3 KB (427 words) - 22:16, 5 June 2020
...al method for obtaining simple groups as groups of automorphisms of simple Lie algebras was discovered by C. Chevalley [[#References|[2]]] (cf. [[Chevalle
...valign="top">[3]</TD> <TD valign="top"> R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972)</TD></TR></table>
1 KB (187 words) - 21:05, 15 November 2017
...he theory of Lie groups and algebraic groups (cf. [[Lie group, semi-simple|Lie group, semi-simple]]).
...simple. There are examples of simple groups that are not absolutely simple and of simple groups that are not strictly simple.
3 KB (510 words) - 17:22, 14 October 2014
...mal subgroups other than $ G $ . A connected Lie group is semi-simple if and only if it splits into a locally direct product of simple non-Abelian norma
...fication of the Lie groups $ G $ that correspond to a given semi-simple Lie algebra $ \mathfrak g $ .
11 KB (1,458 words) - 18:15, 12 December 2019
This page is a copy of the article [[Lie group, semi-simple]] in order to test [[User:Maximilian_Janisch/latexlist|a
...d normal subgroups other than $k$. A connected Lie group is semi-simple if and only if it splits into a locally direct product of simple non-Abelian norma
10 KB (1,401 words) - 13:44, 17 October 2019
...] of a group). The term "solvable group" arose in [[Galois theory|Galois theory]] in connection with the solvability of algebraic equations by radicals.
...into two relatively prime factors there exists a subgroup of order $n_1$, and any two subgroups of order $n_1$ are conjugate. If the order of a finite gr
3 KB (443 words) - 18:25, 26 October 2014
$#C+1 = 204 : ~/encyclopedia/old_files/data/L058/L.0508430 Lie algebra, graded
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20 KB (2,828 words) - 19:11, 26 March 2023
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Generalizations of the hypergeometric functions $ { {} _ {p} F _ {q} } $
4 KB (590 words) - 11:05, 7 June 2020
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...oth a bracket and an associative multiplication defined on the same space, and to impose a "Leibniz rule" relating both operations, stating that the adj
9 KB (1,453 words) - 15:06, 29 July 2024
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and satisfying the condition $ f ( hg ) = \rho ( h) f ( g ) $
13 KB (1,974 words) - 22:12, 5 June 2020
...tric|Hyperbolic metric]]), $M_3$ is the Euclidean Hantzche–Wendt manifold, and $M_2$ is the [[Lens space|lens space]] $L(5,2)$ (see [[#References|[a2]]]).
...the three-dimensional sphere $S^3$, discovered by H.M. Hilden, M.T. Lozano and J.M. Montesinos [[#References|[a3]]]. In fact,
5 KB (677 words) - 13:19, 26 March 2023
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...\mathbf{C} \backslash \{ 0 , \infty \}$ which have algebraic poles at $0$ and $\infty$. A basis of the algebra is given by the set
9 KB (1,398 words) - 10:21, 11 November 2023
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is a real Lie group containing no non-trivial connected complex subgroups. If $ D $
5 KB (656 words) - 03:11, 4 January 2022
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A set equipped with both the algebraic structure of a semi-group and the structure of a topological [[Hausdorff space|Hausdorff space]], such th
16 KB (2,287 words) - 14:36, 19 March 2023
...ction|differentiable]] at $x_0$. Then $fg$ is also differentiable at $x_0$ and
* $x_0\in U$ open subset of $\mathbb R^n$ and the maps $f,g: U \to \mathbb R$ have both the [[Partial derivative|partial
5 KB (757 words) - 10:34, 11 December 2013
...pecially useful for generalizations of the Galois theory of finite, normal and separable field extensions.
...F$ is generated by $P$ and $G$; indeed, it is the [[cross product]] of $P$ and $G$.
3 KB (470 words) - 19:10, 9 November 2016