Trigonalizable element
triangular element
A trigonalizable element of the algebra 
of endomorphisms of a finite-dimensional vector space 
over a field 
is an element 
all eigenvalues of which belong to 
. If 
is algebraically closed, then every element of 
is trigonalizable. For a trigonalizable element 
(and only for such an element) there exists a basis in 
with respect to which the matrix of the endomorphism 
is triangular (or, what is the same, there exists a complete flag in 
that is invariant with respect to 
). A trigonalizable element has a Jordan decomposition over 
. There exist a number of generalizations of the notion of a trigonalizable element in 
for the case that 
is infinite-dimensional (see ).
A trigonalizable element of a finite-dimensional algebra 
over a field 
is an element 
such that the operator of right (or left, depending on the case under consideration) multiplication by 
is a trigonalizable element in the algebra 
. If 
is isomorphic to the algebra 
for some finite-dimensional vector space 
over 
, then these two (formally distinct) definitions reduce to the same concept.
In Lie algebras, trigonalizability of an element 
means trigonalizability of the endomorphism 
(where 
). 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 
, the simple Lie algebra of real matrices of order 2 with trace 0). However, in the case of a solvable algebra 
, this set is even a characteristic ideal of 
.
A trigonalizable element in a connected finite-dimensional Lie group 
is an element 
such that 
is a trigonalizable element in 
(here 
is the adjoint representation of the Lie group 
in the group 
of automorphisms of its Lie algebra 
). If 
is the exponential mapping and 
is a trigonalizable element (in the sense of 2)), then 
is a trigonalizable element of 
. 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 |
Trigonalizable element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trigonalizable_element&oldid=17777