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