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The Lie algebra of an algebraic subgroup (see [[Algebraic group|Algebraic group]]) of the [[General linear group|general linear group]] of all automorphisms of a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583801.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583802.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583803.png" /> is an arbitrary subalgebra of the Lie algebra of all endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583804.png" />, there is a smallest algebraic Lie algebra containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583805.png" />; it is called the algebraic envelope (or hull) of the Lie subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583806.png" />. For a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583807.png" /> over an arbitrary algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583808.png" /> to be algebraic it is necessary that together with every linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583809.png" /> its semi-simple and nilpotent components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838011.png" /> should lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838012.png" /> (see [[Jordan decomposition|Jordan decomposition]]). This condition determines the so-called almost-algebraic Lie algebras. However, it is not sufficient in order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838013.png" /> be an algebraic Lie algebra. In the case of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838014.png" /> of characteristic 0, a necessary and sufficient condition for a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838015.png" /> to be algebraic is that, together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838017.png" />, all operators of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838018.png" /> should lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838020.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838021.png" />-linear mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838023.png" />. The structure of an algebraic algebra has been investigated
+
The Lie algebra of an algebraic subgroup (see
 +
[[Algebraic group|Algebraic group]]) of the
 +
[[General linear group|general linear group]] of all automorphisms of
 +
a finite-dimensional vector space $V$ over a field $k$. If $\mathfrak g$ is an
 +
arbitrary subalgebra of the Lie algebra of all endomorphisms of $V$,
 +
there is a smallest algebraic Lie algebra containing $\mathfrak g$; it is called
 +
the algebraic envelope (or hull) of the Lie subalgebra $\mathfrak g$. For a Lie
 +
algebra $\mathfrak g$ over an arbitrary algebraically closed field $k$ to be
 +
algebraic it is necessary that together with every linear operator $u\in{\mathfrak g}$
 +
its semi-simple and nilpotent components $s$ and $n$ should lie in ${\mathfrak g}$
 +
(see [[Jordan decomposition|Jordan decomposition]]). This condition
 +
determines the so-called almost-algebraic Lie algebras. However, it is
 +
not sufficient in order that ${\mathfrak g}$ be an algebraic Lie algebra. In the
 +
case of a field $k$ of characteristic 0, a necessary and sufficient
 +
condition for a Lie algebra ${\mathfrak g}$ to be algebraic is that, together with
 +
$n$ and $s={\rm diag}(s_1,\dots,s_m)$, all operators of the form $\Phi(s) = {\rm diag}(\Phi(s_1),\dots,\Phi(s_m))$ should lie in ${\mathfrak g}$, where $\Phi$ is an arbitrary $\mathbb Q$-linear mapping from $k$ into $k$. The
 +
structure of an algebraic algebra has been investigated in the case of a field of characteristic $p>0$.
  
in the case of a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838024.png" />.
+
A Lie algebra $L$ over a commutative ring $k$ in which for any element
 
+
$x\in L$ the adjoint transformation ${\rm ad}\; x:y\to [x,y]$ defined on $L$ is the root of some
A Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838025.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838026.png" /> in which for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838027.png" /> the adjoint transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838028.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838029.png" /> is the root of some polynomial with leading coefficient 1 and remaining coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838030.png" />. A finite-dimensional Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838031.png" /> is an algebraic Lie algebra. The converse is false: Over any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838032.png" /> there are infinite-dimensional algebraic Lie algebras with finitely many generators [[#References|[4]]]. A number of questions about algebraic Lie algebras have been solved in the class of nil Lie algebras (cf. [[Lie algebra, nil|Lie algebra, nil]]).
+
polynomial with leading coefficient 1 and remaining coefficients from
 +
$k$. A finite-dimensional Lie algebra over a field $k$ is an algebraic
 +
Lie algebra. The converse is false: Over any field $k$ there are
 +
infinite-dimensional algebraic Lie algebras with finitely many
 +
generators
 +
[[#References|[4]]]. A number of questions about algebraic Lie
 +
algebras have been solved in the class of nil Lie algebras (cf.
 +
[[Lie algebra, nil|Lie algebra, nil]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel,   "Linear algebraic groups" , Benjamin (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Chevalley,   "Théorie des groupes de Lie" , '''2''' , Hermann (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.B. Seligman,   "Modular Lie algebras" , Springer (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.S. Golod,   "On nil algebras and residually finite groups" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' : 2 (1964) pp. 273–276 (In Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD
 
+
valign="top"> A. Borel, "Linear algebraic groups" , Benjamin
 
+
(1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
 +
C. Chevalley, "Théorie des groupes de Lie" , '''2''' , Hermann (1951) {{MR|0051242}} {{ZBL|0054.01303}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
 +
G.B. Seligman, "Modular Lie algebras" , Springer (1967) {{MR|0245627}} {{ZBL|0189.03201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
 +
E.S. Golod, "On nil algebras and residually finite groups", ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' : 2 (1964) pp. 273–276 (in Russian) {{MR|}} {{ZBL|}} </TD></TR></table>
  
 
====Comments====
 
====Comments====
Line 14: Line 40:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.P. Hochschild,   "Basic theory of algebraic groups and Lie algebras" , Springer (1981)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD
 +
valign="top"> G.P. Hochschild, "Basic theory of algebraic groups and
 +
Lie algebras" , Springer (1981) {{MR|0620024}} {{ZBL|0589.20025}} </TD></TR></table>

Latest revision as of 04:43, 4 January 2022

The Lie algebra of an algebraic subgroup (see Algebraic group) of the general linear group of all automorphisms of a finite-dimensional vector space $V$ over a field $k$. If $\mathfrak g$ is an arbitrary subalgebra of the Lie algebra of all endomorphisms of $V$, there is a smallest algebraic Lie algebra containing $\mathfrak g$; it is called the algebraic envelope (or hull) of the Lie subalgebra $\mathfrak g$. For a Lie algebra $\mathfrak g$ over an arbitrary algebraically closed field $k$ to be algebraic it is necessary that together with every linear operator $u\in{\mathfrak g}$ its semi-simple and nilpotent components $s$ and $n$ should lie in ${\mathfrak g}$ (see Jordan decomposition). This condition determines the so-called almost-algebraic Lie algebras. However, it is not sufficient in order that ${\mathfrak g}$ be an algebraic Lie algebra. In the case of a field $k$ of characteristic 0, a necessary and sufficient condition for a Lie algebra ${\mathfrak g}$ to be algebraic is that, together with $n$ and $s={\rm diag}(s_1,\dots,s_m)$, all operators of the form $\Phi(s) = {\rm diag}(\Phi(s_1),\dots,\Phi(s_m))$ should lie in ${\mathfrak g}$, where $\Phi$ is an arbitrary $\mathbb Q$-linear mapping from $k$ into $k$. The structure of an algebraic algebra has been investigated in the case of a field of characteristic $p>0$.

A Lie algebra $L$ over a commutative ring $k$ in which for any element $x\in L$ the adjoint transformation ${\rm ad}\; x:y\to [x,y]$ defined on $L$ is the root of some polynomial with leading coefficient 1 and remaining coefficients from $k$. A finite-dimensional Lie algebra over a field $k$ is an algebraic Lie algebra. The converse is false: Over any field $k$ there are infinite-dimensional algebraic Lie algebras with finitely many generators [4]. A number of questions about algebraic Lie algebras have been solved in the class of nil Lie algebras (cf. Lie algebra, nil).

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) MR0051242 Zbl 0054.01303
[3] G.B. Seligman, "Modular Lie algebras" , Springer (1967) MR0245627 Zbl 0189.03201
[4] E.S. Golod, "On nil algebras and residually finite groups", Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 273–276 (in Russian)

Comments

References

[a1] G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) MR0620024 Zbl 0589.20025
How to Cite This Entry:
Lie algebra, algebraic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_algebraic&oldid=14719
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article