View source for User:Maximilian Janisch/latexlist/Algebraic Groups/Cartan subalgebra

{{TEX|partial}}<div class="Vorlage_Achtung" style="border: 0.18em solid #FF6666; border-left:1em solid #FF6666; margin:0.5em 0em; overflow:hidden; padding:0.5em; text-align: left;">
This page is a copy of the article [[Cartan subalgebra]] in order to test [[User:Maximilian_Janisch/latexlist|automatic LaTeXification]]. This article is not my work.
</div>


''of a finite-dimensional Lie algebra $8$ over a field $k$''

A nilpotent subalgebra of $8$ which is equal to its normalizer in $8$. For example, if $8$ is the Lie algebra of all complex square matrices of a fixed order, then the subalgebra of all diagonal matrices is a Cartan subalgebra in $8$. A Cartan subalgebra can also be defined as a nilpotent subalgebra $1$ in $8$ which is equal to its Fitting null-component (cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]])

\begin{equation} \mathfrak { g } 0 = \{ X \in \mathfrak { g } : \forall H \in \mathfrak { t } \exists \mathfrak { n } X , H \in Z ( ( \text { ad } H ) ^ { n } X , H ( X ) = 0 ) \} \end{equation}

where $a d$ denotes the adjoint representation (cf. [[Lie algebra|Lie algebra]]) of $8$.

Suppose further that $k$ is of characteristic zero. Then for any regular element $x \in \mathfrak { Q }$, the set $n ( X , g )$ of all elements of $8$ which are annihilated by powers of $\operatorname { ad } X$ is a Cartan subalgebra of $8$, and every Cartan subalgebra of $8$ has the form $n ( X , g )$ for some suitable regular element $x$. Each regular element belongs to one and only one Cartan subalgebra. The dimension of all the Cartan subalgebras of $8$ are the same and are equal to the rank of $8$. The image of a Cartan subalgebra under a surjective homomorphism of Lie algebras is a Cartan subalgebra. If $k$ is algebraically closed, then all Cartan subalgebras of $8$ are conjugate; more precisely, they can be transformed into another by operators of the algebraic group $\Omega$ of automorphisms of $8$ whose Lie algebra is the commutator subalgebra of $g$. If $8$ is solvable, then the above assertion holds without the hypothesis that $k$ be algebraically closed.

Let $k$ be either a connected linear algebraic group over an algebraically closed field $k$ of characteristic zero, or a connected Lie group, and let $8$ be its Lie algebra. Then a subalgebra $1$ of $8$ is a Cartan subalgebra if and only if it is the Lie algebra of a [[Cartan subgroup|Cartan subgroup]] of $k$.

Let $8$ be a subalgebra of the Lie algebra $\mathfrak { d } ( V )$ of all endomorphisms of a finite-dimensional vector space $V$ over $k$, and let $8$ be the smallest algebraic Lie algebra in $\mathfrak { d } ( V )$ containing $8$ (cf. [[Lie algebra, algebraic|Lie algebra, algebraic]]). If $1$ is a Cartan subalgebra of $8$, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020550/c02055045.png"/> is a Cartan subalgebra of $8$, and if $1$ is a Cartan subalgebra of $8$ and $1$ is the smallest algebraic subalgebra of $\mathfrak { d } ( V )$ containing $1$, then $1$ is a Cartan subalgebra of $8$ and $t = \overline { t } \cap g$.

Let $k \subset K$ be a field extension. A subalgebra $1$ of $8$ is a Cartan subalgebra if and only if $t \otimes _ { k } K$ is a Cartan subalgebra of $g \otimes k ^ { K }$.

Cartan subalgebras play an especially important role when $8$ is a semi-simple Lie algebra (this was used by E. Cartan [[#References|[1]]]). In this case, every Cartan subalgebra $1$ of $8$ is Abelian and consists of semi-simple elements (see [[Jordan decomposition|Jordan decomposition]]), and the restriction of the [[Killing form|Killing form]] to $1$ is non-singular.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top">  E. Cartan,   "Sur la structure des groupes de transformations finis et continus" , Paris  (1894)  {{MR|}}  {{ZBL|25.0638.02}} </td></tr><tr><td valign="top">[2]</td> <td valign="top">  N. Jacobson,   "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))  {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </td></tr><tr><td valign="top">[3]</td> <td valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)  {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </td></tr><tr><td valign="top">[4]</td> <td valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Ecole Norm. Sup.  (1955)  {{MR|}} {{ZBL|0068.02102}} </td></tr></table>



====Comments====
An element $h \in \mathfrak { q }$ is called regular if the dimension of the Fitting null-component of the endomorphism $h$ of $8$ is minimal.  "Almost-all"  elements of $8$ are regular in the sense that the condition of being regular defines a Zariski-open subset. The result that the Fitting null-component of $h$ for $h$ regular is a Cartan subalgebra holds for finite-dimensional Lie algebras over any infinite field [[#References|[a4]]], p. 59.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)  {{MR|0682756}} {{ZBL|0319.17002}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)  pp. §5.4  {{MR|0323842}} {{ZBL|0254.17004}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)  {{MR|0218496}} {{ZBL|0132.27803}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N. Jacobson,   "Lie algebras" , Dover, reprint  (1979)  ((also: Dover, reprint, 1979))  {{MR|0559927}} {{ZBL|0333.17009}} {{ZBL|0215.38701}} {{ZBL|0144.27103}} {{ZBL|0121.27601}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} {{ZBL|0198.05404}} {{ZBL|0064.27002}} {{ZBL|0064.03503}} {{ZBL|0046.03402}} {{ZBL|0043.26803}} {{ZBL|0039.02803}} {{ZBL|0063.03015}} {{ZBL|0025.30302}} {{ZBL|0025.30301}} {{ZBL|0022.19801}} {{ZBL|0019.19402}} {{ZBL|0018.10302}} {{ZBL|0017.29203}} {{ZBL|0016.20001}} </td></tr></table>