We follow the rules: i, j, m, n, k denote natural numbers, K denotes a field, and a, λ denote elements of K. Let us consider K, λ, n. The Jordan block of λ and n yields a matrix over K and is defined by the conditions (Def. 1). (Def. 1)(i) len (the Jordan block of λ and n) = n, (ii) width (the Jordan block of λ and n) = n, and (iii) for all i, j such that 〈i, j〉 ∈ the indices of the Jordan bloc...

4. Constructible sets 17 5. Connected components 18 6. Subgroups 20 7. Group actions and linearity versus affine 22 7.1. 24 8. Jordan Decomposition 27 8.1. Recall: linear algebra 27 8.2. Jordan decomposition in linear algebraic groups 30 9. Commutative algebraic groups 35 9.1. Basic decomposition 35 9.2. Diagonalizable groups and tori 35 9.2.1. Galois action 39 9.3. Action of tori on affine var...

We give an alternative proof of the Hasse principle for principal homogeneous spaces de ned over elds of virtual cohomological dimension at most one which is based on a special decomposition of elements in Chevalley groups. 1991 Mathematics Subject Classi cation: Primary 20G10; Secondary 11E72, 12D15.

Recommended reading 2 1. The poster child of CRT: the symmetric group 2 1.1. Our best chance of understanding big bad algebraic structures: representations! 3 1.2. Where is this all going? 6 2. Lie algebras 7 2.1. Favorite Examples 7 2.2. Categories, Functors, and the Universal Enveloping Algebra 9 3. Representations of g: a first try 10 3.1. Dual spaces and Hopf algebras: lessons from Group th...

In this paper, we study and analyze the algebraic structure of the circular cone. We establish a new efficient spectral decomposition, set up the Jordan algebra associated with the circular cone, and prove that this algebra forms a Euclidean Jordan algebra with a certain inner product. We then show that the cone of squares of this Euclidean Jordan algebra is indeed the circular cone itself. The...

Suppose g to be a complex semi-simple Lie algebra. In 1955, Chevalley showed that one can assign to it an essentially unique Z-structure, with structure constants of a simple form, themselves unique up to some choices of sign. An algorithm to compute these is also due implicitly to Chevalley, explained explicitly in [Carter:1972] and more recently in [Cohen-Murray-Taylor:2004]. It is reasonably...

We provide an explicit construction for a class of commutative, non-associative algebras each the simple Chevalley groups simply laced type. Moreover, we equip these with associating bilinear form, which turns them into Frobenius algebras. This includes 3876-dimensional algebra on group type E8 acts by automorphisms. also prove that admit structure (axial) decomposition