The long-root elements in Lie algebras of Chevalley type have been well studied and can be characterized as extremal elements, that is, elements x such that the image of (ad x) lies in the subspace spanned by x. In this paper, assuming an algebraically closed base field of characteristic not 2, we find presentations of the Lie algebras of classical Chevalley type by means of minimal sets of ext...

Let Φ̃ be a root system of type F4, and let G be a group generated by non-trivial subgroups Ar, r ∈ Φ̃, satisfying some generalized Steinberg relations, which are also satisfied by root subgroups corresponding to a Moufang octagon. These relations can be considered as a generalization of the element-wise commutator relations in Chevalley groups. The Steinberg presentation specifies the groups sat...

We extend Yangian double to super (or graded) case and give its Drinfeld generators realization by Gauss decomposition. Quantum algebras(in general meaning, it includes Yangians and quantum affine algebras etc.) are new algebraic structures discovered about ten years ago [1-4], they play important roles in the study of soluble statistical models and quantum fields theory. Quantum universal enve...

Introduction. Let G be a finite group acting on a smooth projective curve X. This induces an action of G on the Jacobian JX of X and thus a decomposition of G of JX up to isogeny. The most prominent example of such a situation is the case of the group G ≃ Z/2Z of two elements. Let π : X → Y = X/G denote the canonical quotient map. The first to notice that JX is isogenous to the product JY × P (...

We propose a ‘geometric Chevalley-Warning’ conjecture, that is a motivic extension of the Chevalley-Warning theorem in number theory. It is equivalent to a particular case of a recent conjecture of F. Brown and O.Schnetz. In this paper, we show the conjecture is true for linear hyperplane arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in P. The las...

Theorem (1) There is an involution C of G satisfying: C(h) = h −1 (h ∈ H); (2) C(g) ∼ g −1 for all semisimple elements g; (3) Any two such involutions are conjugate by an inner automorphism;

( b ) H is maximal n i l p o t e n t and every subgroup of f i n i t e l a d e x . . i s of f i n i t e index i n i t s normalizer. Proof of Theorem 6: ( a ) Map Ga -> #, by x,: t -> x,(t) . This is a r a t i o n a l homomorphism. So s i n c e Ga i s a connected , ~ g e b r a i c group s o i s Xa . Hence G is a lgebra ic and connected.. Let R = r a d G . Since R is so lvable and normal it i s f...

In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose coradicals are isomorphic to dual Radford dimension $4p$ for a prime $p>5$. In particular, we obtain families new examples without the Chevalley property.