Fixed point subalgebras of Weil algebras: from geometric to algebraic questions

Kureš, Miroslav

doi:10.1007/978-3-642-20300-8_11

Cited by 2 publications

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“…We notice that [2] studies how the dwindlability is related to the possessing of a non-trivial torus of the identity component of Aut R A, where A is a Weil algebra. In the survey paper [3] a number of claims concerning the form of subalgebras of fixed points of various Weil algebras are demonstrated.…”

Section: Introductionmentioning

confidence: 99%

Dwindlable R-Algebras

Kureš¹

2014

Math. Appl.

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Abstract. The concept of a dwindlable R-algebra is defined for an arbitrary topological R-algebra A. It is proved that if A is dwindlable, then its subalgebra of fixed points is trivial. It is also demonstrated that for an algebra gradable by radical its dwindlability depends on a dwindlability of its factor. IntroductionIn [5] we introduced the concept of so-called dwindlable Weil algebras. This concept should be more precise and general and this is the aim of this paper. Further, the property "to be dwindlable" implied the triviality of the fixed point subalgebra (with respect to all automorphisms) in the mentioned paper. The question is, does it hold in general? Section 1 is devoted to the recalling of basic concepts, exact definitions and the formulation of problems. We also present some original examples for a clear understanding. The main results are in Section 2. We notice that [2] studies how the dwindlability is related to the possessing of a non-trivial torus of the identity component of Aut R A, where A is a Weil algebra. In the survey paper [3] a number of claims concerning the form of subalgebras of fixed points of various Weil algebras are demonstrated.1. Basic concepts 1.1. On rings and homomorphisms 1.1.1. Polynomial rings. Let R be a commutative ring. The polynomial (in one indeterminate) over a ring R is defined as a map a : N 0 → R whose support is finite. Equivalently, polynomials can be defined as sequences (a 0 , a 1 , a 2 , . . . ) such that all but a finite number of a i 's are zeros. A polynomial a : N 0 → R, a(0) = r 0 , a(1) = r 1 , a(2) = r 2 , . . . , will be denoted by r 0 X 0 + r 1 X 1 + r 2 X 2 + . . . (only a finite number of r i 's are non-zeros) with well established simplifications (we do not write terms 0 R X i , etc.); we also define the addition and the multiplication of polynomials in the usual way. Then polynomials form a ring denoted by R[X], MSC (2010): primary 13J20, 13J30.

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Section: Introductionmentioning

confidence: 99%

Dwindlable R-Algebras

Kureš¹

2014

Math. Appl.

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“…Thus, the fundamental problem is a classification of algebras having SA nontrivial. See [4], [5] for related geometric questions and the survey paper [8] for known results up to now, especially for a number of claims concerning the form of subalgebras of fixed points of various Weil algebras.…”

Section: Introductionmentioning

confidence: 99%