Related papers: Chevalley groups over Laurent polynomial rings

As the first main result of this article, we prove that if $e$ and $e'$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus…

In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ for the three types of pair $(\Phi,\Delta)$ (the root system and its subsystem) such that…

We consider the Chevalley involution in the context of real reductive groups. We show that if G(R) is the real points of a connected reductive group, there is an involution, unique up to conjugacy by G(R), taking any semisimple element to a…

Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in…

If $A$ is a unital associative ring and $\ell \geq 2$, then the general linear group $\mathrm{GL}(\ell, A)$ has root subgroups $U_\alpha$ and Weyl elements $n_\alpha$ for $\alpha$ from the root system of type $\mathsf A_{\ell - 1}$.…

Consider a reductive linear algebraic group $G$ acting linearly on a polynomial ring $S$ over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the…

Let $R$ be a ring with unity. The \emph{idempotent graph} $G_{\text{Id}}(R)$ of a ring $R$ is an undirected simple graph whose vertices are the set of all the elements of ring $R$ and two vertices $x$ and $y$ are adjacent if and only if…

For a finite group acting on a polynomial ring, the Chevalley-Shephard-Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. In recent years, progress was…

In this paper we define odd dimensional unitary groups $U_{2n+1}(R,\Delta)$. These groups contain as special cases the odd dimensional general linear groups $GL_{2n+1}(R)$ where $R$ is any ring, the odd dimensional orthogonal and symplectic…

We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in…

This paper has two main parts. In the first part we develop an elementary coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$ is finitely…

Given a finite commutative semigroup $\mathcal{S}$ (written additively), denoted by ${\rm D}(\mathcal{S})$ the Davenport constant of $\mathcal{S}$, namely the least positive integer $\ell$ such that for any $\ell$ elements…

We prove the centrality of $\mathrm{K}_2 (\mathsf{F}_4, \,R)$ for an arbitrary commutative ring $R$. This completes the proof of the centrality of $\mathrm K_2(\Phi,\, R)$ for any root system $\Phi$ of rank $\geq 3$. Our proof uses only…

Rational discrete cohomology and homology for a totally disconnected locally compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$ identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow to…

We prove the centrality of the congruence kernel for the elementary subgroup of a Chevalley group G of rank >1 over an arbitrary noetherian ring R (under some minor restrictions on R if G is of type C_n or G_2).

Let K be an algebraically closed field of characteristic zero, G_m=(K\{0},*) be its multiplicative group, and G_a=(K,+) be its additive group. Consider a commutative linear algebraic group G=G_m^r\times G_a. We study equivariant…

Using the well-known realisation of the root system $ H_4 $ as a folding of $ E_8 $, one can construct examples of $ H_4 $-graded groups from Chevalley groups of type $ E_8 $. Such Chevalley groups are defined over a commutative ring $ R $,…

I present a construction "a` la Chevalley" of affine supergroups associated with simple Lie superalgebras of (classical) type D(2,1;a), for any possible value of the parameter a - in particular, including non-integral values of a. This…

In this paper we study the $\mathbb{A}^1$-invariance of the unstable functor $\mathrm{K}_2(\Phi, R)$ in the case when $\Phi$ is an irreducible root system of type $\mathsf{ADE}$ containing $\mathsf{A}_4$ and not of type $\mathsf{E}_8$. We…

We prove a reduced version of the Chevalley restriction conjecture on the commuting scheme posed by T.H. Chen and B.C. Ng\^o, extending the results of Hunziker for classical groups. In particular, we prove that for any connected reductive…