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Difference between revisions of "Lie algebra, algebraic"

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algebraic it is necessary that together with every linear operator $u\in{\mathfrak g}$
 
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 ${\mathfrak g}$
 
its semi-simple and nilpotent components $s$ and $n$ should lie in ${\mathfrak g}$
(see
+
(see [[Jordan decomposition|Jordan decomposition]]). This condition
[[Jordan decomposition|Jordan decomposition]]). This condition
 
 
determines the so-called almost-algebraic Lie algebras. However, it is
 
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
 
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
 
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
 
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
+
$n$ 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
$\Phi$ 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 $p>0$.
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
 
A Lie algebra $L$ over a commutative ring $k$ in which for any element
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<table><TR><TD valign="top">[1]</TD> <TD
 
<table><TR><TD valign="top">[1]</TD> <TD
 
valign="top"> A. Borel, "Linear algebraic groups" , Benjamin
 
valign="top"> A. Borel, "Linear algebraic groups" , Benjamin
(1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
+
(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
+
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">
(1951)</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">
G.B. Seligman, "Modular Lie algebras" , Springer
+
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>
(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>
 
 
 
 
 
  
 
====Comments====
 
====Comments====
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<table><TR><TD valign="top">[a1]</TD> <TD
 
<table><TR><TD valign="top">[a1]</TD> <TD
 
valign="top"> G.P. Hochschild, "Basic theory of algebraic groups and
 
valign="top"> G.P. Hochschild, "Basic theory of algebraic groups and
Lie algebras" , Springer (1981)</TD></TR></table>
+
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=19573
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article