Classification Of Involutions Research Articles

The classification of involutions and symmetric spaces of modular groups

The involutions and the symmetric spaces associated to the family of modular groups of order 2m are explored. We begin by analyzing the structure of the automorphism group and by establishing which automorphisms are involutions. We conclude by calculating the fixed-point group and symmetric spaces determined by each involution.

Classification of involutions on graded-division simple real algebras

We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.

Classification of involutions on finitary incidence algebras

Let X be a connected partially ordered set and let K be a field of characteristic different from 2. We present necessary and sufficient conditions for two involutions on the finitary incidence algebra of X over K, FI (X), to be equivalent in the case when every multiplicative automorphism of FI (X) is inner. To get the classification of involutions we extend the concept of multiplicative automorphism to finitary incidence algebras and prove the Decomposition Theorem of involutions of [Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra 60 (2012) 181–188] for finitary incidence algebras.

Isomorphy classes of k-involutions of G2

Isomorphy classes of k-involutions have been studied for their correspondence with symmetric k-varieties, also called generalized symmetric spaces. A symmetric k-variety of a k-group G is defined as Gk/Hk where θ : G → G is an automorphism of order 2 that is defined over k and Gk and Hk are the k-rational points of G and H = Gθ, the fixed point group of θ, respectively. This is a continuation of papers written by A. G. Helminck and collaborators [Involutions of SL (2, k), (n > 2), Acta Appl. Math.90(1–2) (2006) 91–119, Classification of involutions of SO (n; k; b), to appear, On the classification of k-involutions, Adv. Math.153(1) (1988) 1–117, Classification of involutions of SL (2, k), Comm. Algebra30(1) (2002) 193–203] expanding on his combinatorial classification over certain fields. Results have been achieved for groups of type A, B and D. Here we begin a series of papers doing the same for algebraic groups of exceptional type.

Limiting cases of Boardman's five halves theorem

AbstractThe famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and ifis the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).

Classification of Involutions on Incidence Algebras

For the case when a finite partially ordered set X has an element that is comparable with all the elements of X, we present necessary and sufficient conditions for two involutions on the incidence algebra of X (over a field of characteristic different from 2) to be equivalent via inner automorphism. Furthermore, we use this classification to get the general classification of involutions.

Involutions onS6with 3–dimensional fixed point set

In this article, we classify all involutions on S^6 with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy CP^3's.