arXiv subjects

Maneesh Thakur

Publications and source records attributed to Maneesh Thakur.

9 recordsLinked to original sources

arXiv2026

Homogeneous Freudenthal algebras and the first Tits construction

Freudenthal algebras over a field are basically the same as Jordan algebras of degree $3$ remaining simple under all base field extensions. These algebras are intimately linked, via their automorphism groups and structure groups, to simple algebraic groups over arbitrary fields. Our main concern here will be the question of when these algebras are homogeneous in the sense that all their Jordan isotopes are isomorphic. We answer this question by presenting various necessary and sufficient conditions for homogeneity and by connecting it with the first Tits construction of cubic Jordan algebras, most notably through investigating Freudenthal division algebras over complete fields under a discrete valuation. We also study the first Tits construction in its own right by producing a local version of it, deriving a local-global principle, and by connecting it with the embeddibility of certain rank-$2$-tori into the automorphism group scheme of an Albert division algebra.

math.RA↗

arXiv2021

Albert algebras and the Tits-Weiss conjecture

We prove the Tits-Weiss conjecture for Albert division algebras over fields of arbitrary characteristics in the affirmative. The conjecture predicts that every norm similarity of an Albert division algebra is a product of a scalar homothety and $U$-operators. This conjecture is equivalent to the Kneser-Tits conjecture for simple, simply connected algebraic groups with Tits index $E^{78}_{8,2}$. We prove that a simple, simply connected algebraic group with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field of arbitrary characteristic, is $R$-trivial, in the sense of Manin, thereby proving the Kneser-Tits conjecture for such groups. The Tits-Weiss conjecture follows as a consequence.

math.GR↗

arXiv2021

On $R$-triviality of $F_4$-II

Simple algebraic groups of type $F_4$ defined over a field $k$ are the full automorphism groups of Albert algebras over $k$. Let $A$ be an Albert algebra over a field $k$ of arbitrary characteristic. We prove that there is an isotope $A^{(v)}$ of $A$ such that the group $\text{\bf Aut}(A^{(v)})$ is $R$-trivial, in the sense of Manin.

math.GR↗

arXiv2019

On $R$-triviality of $F_4$

It is known that simple algebraic groups of type $F_4$ defined over a field $k$ are precisely the full automorphism groups of Albert algebras over $k$. We explore $R$-triviality for the group $\text{\bf Aut}(A)$ when $A$ is an Albert algebra. In this paper, we consider the case when $A$ is an Albert division algebra, that arises from the first Tits construction. We prove that $\text{\bf Aut}(A)$ is $R$-trivial, in the sense of Manin.

math.GR↗

arXiv2019

The cyclicity problem for Albert algebras

In this paper we address the celebrated Albert problem for exceptional Jordan algebras (i.e. Albert algebras): Does every Albert division algebra contain a cubic cyclic subfield? We prove that for any Albert division algebra $A$ over a field $k$ of arbitrary characteristic, there is a suitable isotope that contains a cubic cyclic subfield. It follows from this that for any Albert division algebra $A$ over a field $k$, the structure group $\text{\bf Str}(A)$ always contains a subgroup of type $^3D_4$ defined over $k$.

math.GR↗

arXiv2017

$R$-triviality of some exceptional groups

The main aim of this paper is to prove $R$-triviality for simple, simply connected algebraic groups with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field $k$ of arbitrary characteristic. Let $G$ be such a group. We prove that there exists a quadratic extension $K$ of $k$ such that $G$ is $R$-trivial over $K$, i.e., for any extension $F$ of $K$, $G(F)/R=\{1\}$, where $G(F)/R$ denotes the group of $R$-equivalence classes in $G(F)$, in the sense of Manin (see \cite{M}). As a consequence, it follows that the variety $G$ is retract $K$-rational and that the Kneser-Tits conjecture holds for these groups over $K$. Moreover, $G(L)$ is projectively simple as an abstract group for any field extension $L$ of $K$. In their monograph (\cite{TW}) J. Tits and Richard Weiss conjectured that for an Albert division algebra $A$ over a field $k$, its structure group $Str(A)$ is generated by scalar homotheties and its $U$-operators. This is known to be equivalent to the Kneser-Tits conjecture for groups with Tits index $E_{8,2}^{78}$. We settle this conjecture for Albert division algebras which are first constructions, in affirmative. These results are obtained as corollaries to the main result, which shows that if $A$ is an Albert division algebra which is a first construction and $Γ$ its structure group, i.e., the algebraic group of the norm similarities of $A$, then $Γ(F)/R=\{1\}$ for any field extension $F$ of $k$, i.e., $Γ$ is $R$-trivial.

math.GR↗

arXiv2010

Automorphisms of Albert algebras and a conjecture of Tits and Weiss

Let $k$ be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over $k$ which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra $A$ over a field $k$, every norm similarity of $A$ is inner modulo scalar multiplications. It is known that $k$-forms of $E_8$ with index $E^{78}_{8,2}$ and anisotropic kernel a strict inner $k$-form of $E_6$ correspond bijectively (via Moufang hexagons) to Albert division algebras over $k$. The Kneser-Tits problem for a form of $E_8$ as above is equivalent to the Tits-Weiss conjecture (see \cite{TW}). Hence we provide a solution to the Kneser-Tits problem for forms of $E_8$ arising from pure first Tits construction Albert division algebras. As an application, we prove that for $G={\bf Aut}(A),~G(k)/R=1$, where $A$ is a pure first construction Albert division algebra over $k$ and $R$ stands for $R$-equivalence in the sense of Manin (\cite{M}).

math.GR↗

arXiv2008

Reality Properties of Conjugacy Classes in G_2

Let $G$ be an algebraic group over a field $k$. We call $g\in G(k)$ {\bf real} if $g$ is conjugate to $g^{-1}$ in $G(k)$. In this paper we study reality for groups of type $G_2$ over fields of characteristic different from 2. Let $G$ be such a group over $k$. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in $G(k)$ is real if and only if it is a product of two involutions in $G(k)$. Every unipotent element in $G(k)$ is a product of two involutions in $G(k)$. We discuss reality for $G_2$ over special fields and construct examples to show that reality fails for semisimple elements in $G_2$ over $\Q$ and $\Q_p$. We show that semisimple elements are real for $G_2$ over $k$ with $cd(k)\leq 1$. We conclude with examples of nonreal elements in $G_2$ over $k$ finite, with characteristic $k$ not 2 or 3, which are not semisimple or unipotent.

math.GR↗

arXiv2008

Reality Properties of Conjugacy Classes in algebraic Groups

Let $G$ be an algebraic group defined over a field $k$. We call $g\in G$ {\bf real} if $g$ is conjugate to $g^{-1}$ and $g\in G(k)$ as {\bf $k$-real} if $g$ is real in $G(k)$. An element $g\in G$ is {\bf strongly real} if $\exists h\in G$, $h^{2}=1$ (i.e. $h$ is an {\bf involution}) such that $hgh^{-1}=g^{-1}$. Clearly, strongly real elements are real and are product of two involutions. Let $G$ be a connected adjoint semisimple group over a perfect field $k$, with -1 in the Weyl group. We prove that any strongly regular $k$-real element in $G(k)$ is strongly $k$-real (i.e. is a product of two involutions in $G(k)$). For classical groups, with some mild exceptions, over an arbitrary field $k$ of characteristic not 2, we prove that $k$-real semisimple elements are strongly $k$-real. We compute an obstruction to reality and prove some results on reality specific to fields $k$ with $cd(k)\leq 1$. Finally, we prove that in a group $G$ of type $G_2$ over $k$, characteristic of $k$ different from 2 and 3, any real element in $G(k)$ is strongly $k$-real. This extends our results in \cite{st}, on reality for semisimple and unipotent real elements in groups of type $G_2$.

math.GR↗