Namespaces
Variants
Actions

Difference between revisions of "Trigonalizable element"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex done)
 
Line 1: Line 1:
 +
{{TEX|done}}
 +
 
''triangular element''
 
''triangular element''
  
A trigonalizable element of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942001.png" /> of endomorphisms of a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942002.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942003.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942004.png" /> all eigenvalues of which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942005.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942006.png" /> is algebraically closed, then every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942007.png" /> is trigonalizable. For a trigonalizable element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942008.png" /> (and only for such an element) there exists a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t0942009.png" /> with respect to which the matrix of the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420010.png" /> is triangular (or, what is the same, there exists a complete flag in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420011.png" /> that is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420012.png" />). A trigonalizable element has a [[Jordan decomposition|Jordan decomposition]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420013.png" />. There exist a number of generalizations of the notion of a trigonalizable element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420014.png" /> for the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420015.png" /> is infinite-dimensional (see ).
+
A trigonalizable element of the algebra $  \mathop{\rm End}\nolimits \  V $
 +
of endomorphisms of a finite-dimensional vector space $  V $
 +
over a field $  k $
 +
is an element $  X \in  \mathop{\rm End}\nolimits \  V $
 +
all eigenvalues of which belong to $  k $.  
 +
If $  k $
 +
is algebraically closed, then every element of $  \mathop{\rm End}\nolimits \  V $
 +
is trigonalizable. For a trigonalizable element $  X $(
 +
and only for such an element) there exists a basis in $  V $
 +
with respect to which the matrix of the endomorphism $  X $
 +
is triangular (or, what is the same, there exists a complete flag in $  V $
 +
that is invariant with respect to $  X $).  
 +
A trigonalizable element has a [[Jordan decomposition|Jordan decomposition]] over $  k $.  
 +
There exist a number of generalizations of the notion of a trigonalizable element in $  \mathop{\rm End}\nolimits \  V $
 +
for the case that $  V $
 +
is infinite-dimensional (see ).
 +
 
 +
A trigonalizable element of a finite-dimensional algebra  $  A $
 +
over a field  $  k $
 +
is an element  $  a \in A $
 +
such that the operator of right (or left, depending on the case under consideration) multiplication by  $  a $
 +
is a trigonalizable element in the algebra  $  \mathop{\rm End}\nolimits _{k} \  A $.  
 +
If  $  A $
 +
is isomorphic to the algebra  $  \mathop{\rm End}\nolimits \  V $
 +
for some finite-dimensional vector space  $  V $
 +
over  $  k $,
 +
then these two (formally distinct) definitions reduce to the same concept.
  
A trigonalizable element of a finite-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420016.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420017.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420018.png" /> such that the operator of right (or left, depending on the case under consideration) multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420019.png" /> is a trigonalizable element in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420021.png" /> is isomorphic to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420022.png" /> for some finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420023.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420024.png" />, then these two (formally distinct) definitions reduce to the same concept.
+
In Lie algebras, trigonalizability of an element $  x \in A $
 +
means trigonalizability of the endomorphism  $  \mathop{\rm ad}\nolimits _{x} $(
 +
where  $  \mathop{\rm ad}\nolimits _{x} (y) = [x,\  y] $).  
 +
The set of all trigonalizable elements in a Lie algebra is, in general, not closed with respect to the operations of addition and commutation (for example, for  $  \mathfrak{ gl } (2,\  \mathbf R ) $,
 +
the simple Lie algebra of real matrices of order 2 with trace 0). However, in the case of a solvable algebra  $  A $,
 +
this set is even a characteristic ideal of  $  A $.
  
In Lie algebras, trigonalizability of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420025.png" /> means trigonalizability of the endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420026.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420027.png" />). The set of all trigonalizable elements in a Lie algebra is, in general, not closed with respect to the operations of addition and commutation (for example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420028.png" />, the simple Lie algebra of real matrices of order 2 with trace 0). However, in the case of a solvable algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420029.png" />, this set is even a characteristic ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420030.png" />.
 
  
A trigonalizable element in a connected finite-dimensional Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420031.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420033.png" /> is a trigonalizable element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420034.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420035.png" /> is the adjoint representation of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420036.png" /> in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420037.png" /> of automorphisms of its Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420038.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420039.png" /> is the exponential mapping and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420040.png" /> is a trigonalizable element (in the sense of 2)), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420041.png" /> is a trigonalizable element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094200/t09420042.png" />. The converse is, in general, false.
+
A trigonalizable element in a connected finite-dimensional Lie group $  G $
 +
is an element $  g \in G $
 +
such that $  \mathop{\rm Ad}\nolimits _{g} $
 +
is a trigonalizable element in $  \mathop{\rm End}\nolimits \  \mathfrak g $(
 +
here $  \mathop{\rm Ad}\nolimits : \  G \rightarrow  \mathop{\rm Aut}\nolimits \  \mathfrak g $
 +
is the adjoint representation of the Lie group $  G $
 +
in the group $  \mathop{\rm Aut}\nolimits \  \mathfrak g \subset  \mathop{\rm End}\nolimits \  \mathfrak g $
 +
of automorphisms of its Lie algebra $  \mathfrak g $).  
 +
If $  \mathop{\rm exp}\nolimits : \  \mathfrak g \rightarrow G $
 +
is the exponential mapping and $  X \in \mathfrak g $
 +
is a trigonalizable element (in the sense of 2)), then $  \mathop{\rm exp}\nolimits (X) $
 +
is a trigonalizable element of $  G $.  
 +
The converse is, in general, false.
  
 
Lie algebras and Lie groups all elements of which are trigonalizable are called trigonalizable algebras or groups, respectively, and also supersolvable Lie algebras, respectively (cf. [[Lie algebra, supersolvable|Lie algebra, supersolvable]]; [[Lie group, supersolvable|Lie group, supersolvable]]).
 
Lie algebras and Lie groups all elements of which are trigonalizable are called trigonalizable algebras or groups, respectively, and also supersolvable Lie algebras, respectively (cf. [[Lie algebra, supersolvable|Lie algebra, supersolvable]]; [[Lie group, supersolvable|Lie group, supersolvable]]).

Latest revision as of 12:56, 22 December 2019


triangular element

A trigonalizable element of the algebra $ \mathop{\rm End}\nolimits \ V $ of endomorphisms of a finite-dimensional vector space $ V $ over a field $ k $ is an element $ X \in \mathop{\rm End}\nolimits \ V $ all eigenvalues of which belong to $ k $. If $ k $ is algebraically closed, then every element of $ \mathop{\rm End}\nolimits \ V $ is trigonalizable. For a trigonalizable element $ X $( and only for such an element) there exists a basis in $ V $ with respect to which the matrix of the endomorphism $ X $ is triangular (or, what is the same, there exists a complete flag in $ V $ that is invariant with respect to $ X $). A trigonalizable element has a Jordan decomposition over $ k $. There exist a number of generalizations of the notion of a trigonalizable element in $ \mathop{\rm End}\nolimits \ V $ for the case that $ V $ is infinite-dimensional (see ).

A trigonalizable element of a finite-dimensional algebra $ A $ over a field $ k $ is an element $ a \in A $ such that the operator of right (or left, depending on the case under consideration) multiplication by $ a $ is a trigonalizable element in the algebra $ \mathop{\rm End}\nolimits _{k} \ A $. If $ A $ is isomorphic to the algebra $ \mathop{\rm End}\nolimits \ V $ for some finite-dimensional vector space $ V $ over $ k $, then these two (formally distinct) definitions reduce to the same concept.

In Lie algebras, trigonalizability of an element $ x \in A $ means trigonalizability of the endomorphism $ \mathop{\rm ad}\nolimits _{x} $( where $ \mathop{\rm ad}\nolimits _{x} (y) = [x,\ y] $). The set of all trigonalizable elements in a Lie algebra is, in general, not closed with respect to the operations of addition and commutation (for example, for $ \mathfrak{ gl } (2,\ \mathbf R ) $, the simple Lie algebra of real matrices of order 2 with trace 0). However, in the case of a solvable algebra $ A $, this set is even a characteristic ideal of $ A $.


A trigonalizable element in a connected finite-dimensional Lie group $ G $ is an element $ g \in G $ such that $ \mathop{\rm Ad}\nolimits _{g} $ is a trigonalizable element in $ \mathop{\rm End}\nolimits \ \mathfrak g $( here $ \mathop{\rm Ad}\nolimits : \ G \rightarrow \mathop{\rm Aut}\nolimits \ \mathfrak g $ is the adjoint representation of the Lie group $ G $ in the group $ \mathop{\rm Aut}\nolimits \ \mathfrak g \subset \mathop{\rm End}\nolimits \ \mathfrak g $ of automorphisms of its Lie algebra $ \mathfrak g $). If $ \mathop{\rm exp}\nolimits : \ \mathfrak g \rightarrow G $ is the exponential mapping and $ X \in \mathfrak g $ is a trigonalizable element (in the sense of 2)), then $ \mathop{\rm exp}\nolimits (X) $ is a trigonalizable element of $ G $. The converse is, in general, false.

Lie algebras and Lie groups all elements of which are trigonalizable are called trigonalizable algebras or groups, respectively, and also supersolvable Lie algebras, respectively (cf. Lie algebra, supersolvable; Lie group, supersolvable).

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] B.I. Plotkin, "Groups of automorphisms of algebraic systems" , Wolters-Noordhoff (1972) MR0344322 Zbl 0229.20004
[3] M.M. Postnikov, "Linear algebra and differential geometry" , Moscow (1979) (In Russian) Zbl 0744.51001 Zbl 0491.51005


Comments

References

[a1] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
How to Cite This Entry:
Trigonalizable element. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Trigonalizable_element&oldid=44323
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article