This expository paper discusses a few of the results concerning the structure theory and representation theory of algebraic groups over a fixed algebraically closed field K, such as the linearization of affine groups, the Jordan-Chevalley decomposition, and the Lie-Kolchin theorem.

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.

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

We analyze the structure of the algebra K〈x〉n of symmetric polynomials in non-commuting variables in so far as it relates to K[x]n , its commutative counterpart. Using the “place-action” of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of K〈x〉n analogous to the classical theorems of Chevalley, Sh...

We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. The aim of this paper is to prove a Chevalley restriction theorem and its double analogue for the cyclic quiver. When the quiver is of type Â0, we recover the results for gln. The proof of our Chevalley restriction theorem is similar to the proof for gln; however, the proof of the double analogue uses a theo...

Clifford algebras are naturally associated with quadratic forms. These algebras are Z2 -graded by construction. However, only a Zn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V ) ↔ ∧ V and an ordering, guarantees a multivector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomor...