Cartan Subalgebras in Lie Algebras of Associative Algebras

Siciliano, Salvatore

doi:10.1080/00927870600938928

Cited by 2 publications

(20 citation statements)

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“…As p does not divide n, this implies that / 9 has no multiple root in any extension field of F. Thus, H consists of semisimple elements of FG which commute pairwise. By [9], it follows that T :-(H) F is a torus of FG. Since 0 is the only element of FG which is both nilpotent and semisimple, we have TnRad(FG) = {0}.…”

Section: P R O O Fmentioning

confidence: 95%

“…As usual, for a finite dimensional vector space V and a linear tranformation / of V, we denote by V 0 (f) the null We say that an element x of an associative algebra A over a field F is semisimple if the minimum polynomial of x has no multiple roots in any extension field of F. A torus of A is defined to be an Abelian subalgebra of A consisting of semisimple elements. In [9], the following results about the Cartan subalgebras of ^Lie were obtained: THEOREM 3 . Let A be an associative algebra over a field F.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…(T*), so that (T*) 3 F is a complement of Rad(B) in B, too. By the Wedderburn-Malcev Theorem, T* and T* 9 are conjugate under Rad(B). But (T*) F C Z{B) now forces {T*) F = (T*) 9 F , hence T* 9 = T*.…”

Section: Of Lemma 1 1 + Rad(a) Is Nilpotent Hence C a -(T*) Is Alsmentioning

confidence: 99%

“…By the Wedderburn-Malcev Theorem, T* and T* 9 are conjugate under Rad(B). But (T*) F C Z{B) now forces {T*) F = (T*) 9 F , hence T* 9 = T*. Theorem 1 and the Wedderburn-Malcev Theorem imply the existence of a Carter subgroup of A*.…”

Section: Of Lemma 1 1 + Rad(a) Is Nilpotent Hence C a -(T*) Is Alsmentioning

confidence: 99%

“…Given a unitary associative algebra A of finite dimension over a field, we can consider both the group of units A* of A and, on the other hand, the Lie algebra A Lie associated to A by means of the Lie product (x, y) = xy -yx for every x,y & A. From results obtained in [9] (see Theorem 3 of the next section) we know, in particular, that the Cartan subalgebras of A Lie are associative subalgebras of A. Following [5], we say that A is solvable if A/ Rad(^4) is commutative.…”

Section: Introductionmentioning

confidence: 99%

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Carter subgroups in the group of units of an associative algebra

Bauer¹,

Siciliano²

2005

Bull. Austral. Math. Soc.

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In this paper we examine some properties of the Carter subgroups in the group of units of certain associative algebras. A description of the Carter subgroups in the case of a solvable associative algebra is obtained. Moreover, given an associative algebra A, we study relationships between the Cartan subalgebras of the Lie algebra associated with A and the Carter subgroups of the group of units of A.

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Section: P R O O Fmentioning

confidence: 95%

Section: Preliminaries and Notationsmentioning

confidence: 99%

Section: Of Lemma 1 1 + Rad(a) Is Nilpotent Hence C a -(T*) Is Alsmentioning

confidence: 99%

Section: Of Lemma 1 1 + Rad(a) Is Nilpotent Hence C a -(T*) Is Alsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Carter subgroups in the group of units of an associative algebra

Bauer¹,

Siciliano²

2005

Bull. Austral. Math. Soc.

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About Cartan-Subalgebras in Lie-Algebras associated to Associative Algebras

Wirsing¹

2012

Preprint

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Every associative K-algebra A is with respect to the multiplication a • b := ab − ba for all a, b ∈ A a Lie-Algebra A • , also known as the Lie-Algebra associated with A. In [8] S. Siciliano studies Cartan-Subalgebras of A • . These are nilpotent subalgebras C of A • which coincide with the normalizerFor every finite-dimensional associative unitary K-algebra A Siciliano proofs that the Cartan-Subalgebras of A • are exactly the centralizers of the maximal tori of A. Especially, Cartan-Subalgebras of A • are subalgebras of the associative algebra A. A torus is a commutative unitary subalgebra of A for which every element is separable over K. An element is separable if its minimal polynomial is a product of pairwise irreducible separable polynomials of K[t]. For the algebra classes "finite-dimensional central division algebras" and "finite-dimensional soluble algebras" S.Siciliano describes the Cartan-Subalgebras in terms of 'maximal separable subfields' and 'radical complements'.At first we give alternative proves for Sicilianos results concerning these two algebra classes. As an example we compute the Cartan-Subalgebras for soluble group algebras and especially for soluble group algebras related to dihedral groups. After this we describe the Cartan-Subalgebras for finite-dimensional associative division, simple, semi-simple and reduced algebras.As a associative subalgebra we analyze the associative structure of Cartan-Subalgebras. This investigation is closely connected to the question for which finite-dimensional associative unitary algebras its group of units is nilpotent. We close this article by giving a strategy for computing Cartan-Subalgebras for associative algebras with separable radical complements. An easy consequence is again the description of the Cartan-Subalgebras in the case of

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Cartan Subalgebras in Lie Algebras of Associative Algebras

Cited by 2 publications

References 7 publications

Carter subgroups in the group of units of an associative algebra

Carter subgroups in the group of units of an associative algebra

About Cartan-Subalgebras in Lie-Algebras associated to Associative Algebras

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