Automorphisms of Chevalley groups over commutative rings

Abstract

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems ${\mathbf{A}}_2,{\mathbf{F}}_4,{\mathbf{B}}_l,{\mathbf{C}}_l$; with 1/2 and 1/3 for the system ${\mathbf{G}}_2$) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However, the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description.

KEYWORDS:

• Automorphisms
• Chevalley groups
• commutative rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20G35

1 Introduction

1.1 Automorphisms and isomorphisms of classical linear groups

Automorphisms and isomorphisms of linear groups are studied by mathematicians from the beginning of XX century. First papers on automorphisms and isomorphisms of linear groups appeared already in the beginning of the 20th century. In particular, in the paper by Schreier and van der Warden [54] they described all automorphisms of the group $\text{PSL}{ }_n$ $(\mathit{n⩾}3)$ over an arbitrary field. Later on, Hua [33] generalized this method and applied it to the description of automorphisms of symplectic groups over a field of characteristic $\ne 2$. Diedonne [27] (1951) and Rickart [52] (1950) introduced the involution method, and described automorphisms of the group $\text{GL}{ }_n$ ($\mathit{n⩾}3)$ over a skew field, and then also of unitary and symplectic groups over skew fields of characteristic $\ne 2$ [53].

The first step toward the description of automorphisms of classical groups over rings was made by Hua and Reiner [32]. They dealt with the case $\text{GL}{ }_n(\mathbb{Z})$. This result was extended to non-commutative principal ideal domains by Landin and Reiner in [39] and by Yan Shi-jian in [57].

The methods of the papers mentioned above were based mostly on studying involutions in the corresponding linear groups.

O’Meara in 1976 invented very different (geometrical) method, which did not use involutions. By its aid, O’Meara described automorphisms of the group $\text{GL}{ }_n$ ($\mathit{n⩾}3$) over domains [45] and automorphisms of symplectic groups of a special form over fields (so-called groups rich in transvections) [46]. Independently, Yan Shi-jian in [57] described automorphisms of the group $E_n(R),\mathit{n⩾}3$, where R is a domain of characteristic $\ne 2$ using the involution method. In the paper [43] Pomfret and MacDonald studied automorphisms of the groups $\text{GL}{ }_n,\mathit{n⩾}3$, over a commutative local ring with 1/2. Further on, Waterhouse in [66] obtained a description of automorphisms of the group $\text{GL}{ }_n,\mathit{n⩾}3$, over arbitrary commutative rings with 1/2.

In 1982 Petechuk [47] described automorphisms of the groups $\text{GL}{ }_n, \text{SL}{ }_n$ ($\mathit{n⩾}4$) over arbitrary commutative rings. If n = 3, then automorphisms of linear groups are not always standard [48]. They are standard either if in a ring 2 is invertible, or if a ring is a domain, or it is a semisimple ring.

McQueen and McDonald in [44] obtained the description of automorphisms of the groups $\text{Sp}{ }_n$, $\mathit{n⩾}6$ over commutative local rings with 1/2. Continuing research in this direction, in 1980 Petechuk in [49] studied automorphisms of symplectic groups over arbitrary commutative local rings. In 1982 he extended description of automorphisms to the case $\text{Sp}{ }_n(R),\mathit{n⩾}6$, over arbitrary commutative ring R, using the localization method, see [50].

Isomorphisms of the groups $\text{GL}{ }_n(R)$ and $\text{GL}{ }_m(S)$ over arbitrary associative rings with 1/2 for $n,\mathit{m⩾}3$ were described in 1981 by Golubchik and Mikhalev [29] and independently by Zelmanov [67]. In 1997 Golubchik described isomorphisms between these groups for $n,\mathit{m⩾}4$, over arbitrary associative rings with 1 [30].

In 1983 Golubchik and Mikhalev in [28] studied isomorphisms of unitary linear groups over arbitrary associative rings with 1/2, with some conditions for the dimension of the group and the rank of the form. For the case when $n=2k$ and the hyperbolic rank of the form Q is maximal, the automorphism of $U_n(R,Q),\mathit{k⩾}3$, were independently classified in 1985 by Zelmanov, see [67].

1.2 Automorphisms and isomorphisms of Chevalley groups

In 50-th years of the previous century Chevalley, Steinberg and others introduced the concept of Chevalley groups over commutative rings. The foundations of the theory of Chevalley groups have been laid in the papers of Chevalley, Tits, Borel, Weil, Grothendieck, Demazure, Stenberg, etc. In 1956–1958 Chevalley obtained a classification of semisimple algebraic groups over algebraically closed fields. Later on, Chevalley showed that all semisimple groups over an algebraically closed field are actually defined under $\mathbb{Z}$, or, in other words, are obtained as a result of expanding to an arbitrary ring of some group scheme defined over $\mathbb{Z}$. These group schemes are called Chevalley-Demazure schemes. The groups of points of Chevalley-Demazure schemes over commutative rings are called Chevalley groups. Chevalley groups include classical linear groups (special linear $\text{SL}$, special orthogonal $\text{SO}$, symplectic $\text{Sp}$, spinor $\text{Spin}$, and also projective groups connected with them) over commutative rings. Finite simple groups of Lie type are the central quotients of Chevalley groups.

Isomorphisms and automorphisms of Chevalley groups over different classes of rings were intensively studied. The description of isomorphisms of Chevalley groups over fields was obtained by Steinberg [60] for the finite case and by Humphreys [34] for the infinite one. Many papers are devoted to description of automorphisms of Chevalley groups over commutative rings. We can mention here the papers of Borel–Tits [6], Carter–Chen Yu [19], Chen Yu [20]–[22], Abe [1], Klyachko [38].

Usually complete description of automorphisms of Chevalley groups means standardity of all these automorphisms, that is, all automorphisms are compositions of some simple and well-described types of automorphisms: inner automorphisms, automorphisms induced by ring automorphisms, etc.

Abe in [1] proved the standardity of automorphisms for Noetherian rings with 1/2, which could help to close the question of automorphisms of Chevalley groups over arbitrary commutative rings with 1/2. However, in considering the case of adjoint elementary groups has a gap, which cannot be eliminated by the methods of this article.

The cases when the ring contains a lot of invertible integers (in some sense) are completely clarified in the paper of Klyachko [38].

In the paper [8] Bunina proved that automorphisms of adjoint elementary Chevalley groups with root systems ${\mathbf{A}}_l,{\mathbf{D}}_l,{\mathbf{E}}_l,\mathit{l⩾}2$, over local rings with invertible 2 can be represented as the composition of ring automorphism and an automorphism–conjugation (by automorphism-conjugation we call conjugation of elements of a Chevalley group in the adjoint representation by some matrix from the normalizer of this group in $\text{GL} (V)$). By the similar token it was proved in [10] that every automorphism of an arbitrary Chevalley (or its arbitrary subgroup) group is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. In the same paper it was obtained the theorem describing the normalizer of Chevalley groups in their adjoint representation, which also holds for local rings without 1/2.

In the series of papers [9, 11–13, 17] the similar methods made it possible to obtain the standardity of all automorphisms of Chevalley groups $G(\Phi ,R)$ where $\Phi ={\mathbf{F}}_4,{\mathbf{B}}_l,\mathit{l⩾}3$, R is a local ring and $1/2\in R$, or $\Phi ={\mathbf{G}}_2$ and 1/2, $1/3\in R$. The same is true for $\Phi ={\mathbf{A}}_l,{\mathbf{D}}_l$ ${\mathbf{E}}_l$, ${\mathbf{G}}_2,\mathit{l⩾}2$, R is a local ring and $1/2\notin R$. As we already mentioned the case ${\mathbf{C}}_l$ (symplectic linear groups and projective symplectic linear groups) was considered in the papers of Petechuk and Golubchik–Mikhalev (even for non-commutative rings).

The non-standard automorphisms are described by Steinberg in [59] for the cases of Chevalley groups of types ${\mathbf{B}}_2$ and ${\mathbf{F}}_4$ over fields of characteristic 2 and of type ${\mathbf{G}}_2$ over fields of characteristic 3. For fields of characteristic 2 also there exists an isomorphism between Chevalley groups of types ${\mathbf{B}}_l$ and ${\mathbf{C}}_l$, $\mathit{l⩾}3$. In [48] Petechuk described (non-standard) automorphisms of Chevalley groups of the type ${\mathbf{A}}_2$ over local rings without 1/2. Therefore the cases of Chevalley groups of the types ${\mathbf{A}}_2,{\mathbf{B}}_l,{\mathbf{C}}_l,{\mathbf{F}}_4$ over rings without 1/2 and of the type ${\mathbf{G}}_2$ over rings without 1/3 require separate consideration.

In the paper [14] Bunina used the localization method and ideas of Petechuk and generalized the description of automorphisms of Chevalley groups over local rings to adjoint Chevalley groups over arbitrary commutative rings. In the paper [15] the isomorphisms between these Chevalley groups were described.

In this paper we extend the result of [14] to arbitrary Chevalley groups over rings.

The paper is organized as follows. Section 2 deals with definitions and formulation of the Main Theorem. The proof of the Main Theorem for elementary case is situated in Section 3. The next Section 4 is devoted to the proof of the Main Theorem in the general case.

2 Definitions and main theorem

2.1 Root systems and semisimple Lie algebras

We fix an indecomposable root system $\Phi$ of the rank $\mathcal{l}>1$, with the system of simple roots Δ, the set of positive (negative) roots ${\Phi }^{+}$ (${\Phi }^{-}$), and the Weil group W. Recall that any two roots of the same length are conjugate under the action of the Weil group. Let $\left|{\Phi }^{+}\right|=m$. More detailed texts about root systems and their properties can be found in the books [7, 35].

Recall also that for $\alpha ,\beta \in \Phi$ $$〈\alpha ,\beta 〉=2\frac{(\alpha ,\beta )}{(\beta ,\beta )}.$$

Suppose now that we have a semisimple complex Lie algebra $\mathcal{L}$ with the Cartan subalgebra $\mathcal{H}$ (more details about semisimple Lie algebras can be found, for instance, in the book [35]).

Lie algebra $\mathcal{L}$ has a decomposition $\mathcal{L}=\mathcal{H}\oplus \sum _{\alpha \ne 0}{\mathcal{L}}_{\alpha }$, $${\mathcal{L}}_{\alpha }:=\{x\in \mathcal{L}⏧[h,x]=\alpha (h)x\text{for every }h\in \mathcal{H}\},$$and if ${\mathcal{L}}_{\alpha }\ne 0$, then $\dim {\mathcal{L}}_{\alpha }=1$, all nonzero $\alpha \in \mathcal{H}$ such that ${\mathcal{L}}_{\alpha }\ne 0$, form some root system $\Phi$. The root system $\Phi$ and the semisimple Lie algebra $\mathcal{L}$ over $\mathbb{C}$ uniquely (up to automorphism) define each other.

On the Lie algebra $\mathcal{L}$ we can introduce a bilinear Killing form $\varkappa (x,y)= \text{tr} ( \text{ad} x \text{ad} y),$ that is non-degenerated on $\mathcal{H}$. Therefore we can identify the spaces $\mathcal{H}$ and ${\mathcal{H}}^{*}$.

We can choose a basis $\{h_1,\dots ,h_l\}$ in $\mathcal{H}$ and for every $\alpha \in \Phi$ elements $x_{\alpha }\in {\mathcal{L}}_{\alpha }$ so that $\{h_i;x_{\alpha }\}$ is a basis in $\mathcal{L}$ and for every two elements of this basis their commutator is an integral linear combination of the elements of the same basis. This basis is called a Chevalley basis.

2.2 Elementary Chevalley groups

Introduce now elementary Chevalley groups (see [59]).

Let $\mathcal{L}$ be a semisimple Lie algebra (over $\mathbb{C}$) with a root system $\Phi ,\pi :\mathcal{L}\to \mathfrak{g}\mathfrak{l}(V)$ be its finitely dimensional faithful representation (of dimension n). If $\mathcal{H}$ is a Cartan subalgebra of $\mathcal{L}$, then a functional $\lambda \in {\mathcal{H}}^{*}$ is called a weight of a given representation, if there exists a nonzero vector $v\in V$ (that is called a weight vector) such that for any $h\in \mathcal{H}$ $\pi (h)v=\lambda (h)v.$

In the space V in the Chevalley basis all operators $\pi {(x_{\alpha })}^k/k!$ for $k\in \mathbb{N}$ are written as integral (nilpotent) matrices. An integral matrix also can be considered as a matrix over an arbitrary commutative ring with 1. Let R be such a ring. Consider matrices n × n over R, matrices $\pi {(x_{\alpha })}^k/k!$ for $\alpha \in \Phi ,k\in \mathbb{N}$ are included in $M_n(R)$.

Now consider automorphisms of the free module Rⁿ of the form $$\hspace{0.17em}\exp \hspace{0.17em}(t{x}_{\alpha })=x_{\alpha }(t)=1+\mathit{t\pi }(x_{\alpha })+t^2\pi {(x_{\alpha })}^2/2+\dots +t^k\pi {(x_{\alpha })}^k/k!+\cdots$$

Since all matrices $\pi (x_{\alpha })$ are nilpotent, we have that this series is finite. Automorphisms $x_{\alpha }(t)$ are called elementary root elements. The subgroup in $\text{Aut} (R^n)$, generated by all $x_{\alpha }(t),\alpha \in \Phi ,t\in R$, is called an elementary Chevalley group (notation: $E_{\pi }(\Phi ,R)$).

In elementary Chevalley group we can introduce the following important elements and subgroups:

• $w_{\alpha }(t)=x_{\alpha }(t)x_{-\alpha }(-t^{-1})x_{\alpha }(t),\alpha \in \Phi ,t\in R^{*}$;

• $h_{\alpha }(t)=w_{\alpha }(t)w_{\alpha }{(1)}^{-1}$;

• N is generated by all $w_{\alpha }(t),\alpha \in \Phi ,t\in R^{*}$;

• H is generated by all $h_{\alpha }(t),\alpha \in \Phi ,t\in R^{*}$;

• The subgroup $U=U(\Phi ,R)$ of the Chevalley group $G(\Phi ,R)$ (resp. $E(\Phi ,R)$) is generated by elements $x_{\alpha }(t),\alpha \in {\Phi }^{+},t\in R$, the subgroup $V=V(\Phi ,R)$ is generated by elements $x_{-\alpha }(t),\alpha \in {\Phi }^{+}$ $t\in R$.

The action of $x_{\alpha }(t)$ on the Chevalley basis is described in [18, 65].

It is known that the group N is a normalizer of H in elementary Chevalley group, the quotient group N/H is isomorphic to the Weil group $W(\Phi )$.

All weights of a given representation (by addition) generate a lattice (free Abelian group, where every $\mathbb{Z}$-basis is also a $\mathbb{C}$-basis in ${\mathcal{H}}^{*}$), that is called the weight lattice ${\Lambda }_{\pi }$.

Elementary Chevalley groups are defined not even by a representation of the Chevalley groups, but just by its weight lattice. More precisely, up to an abstract isomorphism an elementary Chevalley group is completely defined by a root system $\Phi$, a commutative ring R with 1 and a weight lattice ${\Lambda }_{\pi }$.

Among all lattices we can mark two: the lattice corresponding to the adjoint representation, it is generated by all roots (the root lattice Λ_ad) and the lattice generated by all weights of all representations (the lattice of weights Λ_sc). For every faithful representation π we have the inclusion ${\Lambda }_{ad}\subseteq {\Lambda }_{\pi }\subseteq {\Lambda }_{sc}.$ Respectively, we have the adjoint and simply connected elementary Chevalley groups.

Every elementary Chevalley group satisfies the following relations:

(R1) $\forall \alpha \in \Phi$ $\forall t,u\in R$ $x_{\alpha }(t)x_{\alpha }(u)=x_{\alpha }(t+u)$;

(R2) $\forall \alpha ,\beta \in \Phi$ $\forall t,u\in R$ $\alpha +\beta \ne 0\Rightarrow$ $$[x_{\alpha }(t),x_{\beta }(u)]=x_{\alpha }(t)x_{\beta }(u)x_{\alpha }(-t)x_{\beta }(-u)=\prod x_{i\alpha +j\beta }(c_{ij}{t}^i{u}^j),$$where i, j are integers, product is taken by all roots $\mathit{i\alpha }+\mathit{j\beta }$, taken in some fixed order; c_ij are integer numbers not depending on t and u, but depending on α and β and the order of roots in the product.

(R3) $\forall \alpha \in \Phi$ $w_{\alpha }=w_{\alpha }(1)$;

(R4) $\forall \alpha ,\beta \in \Phi$ $\forall t\in R^{*}$ $w_{\alpha }{h}_{\beta }(t)w_{\alpha }^{-1}=h_{w_{\alpha }(\beta )}(t)$;

(R5) $\forall \alpha ,\beta \in \Phi$ $\forall t\in R^{*}$ $w_{\alpha }{x}_{\beta }(t)w_{\alpha }^{-1}=x_{w_{\alpha }(\beta )}(\mathit{ct})$, where $c=c(\alpha ,\beta )=\pm 1$;

(R6) $\forall \alpha ,\beta \in \Phi$ $\forall t\in R^{*}$ $\forall u\in R$ $h_{\alpha }(t)x_{\beta }(u)h_{\alpha }{(t)}^{-1}=x_{\beta }(t^{〈\beta ,\alpha 〉}u)$.

For a given $\alpha \in \Phi$ by $X_{\alpha }$ we denote the subgroup $\{x_{\alpha }(t)\mathit{⏧t}\in R\}$.

2.3 Chevalley groups

Introduce now Chevalley groups (see [5, 18, 23, 25, 59, 64, 65], and references therein).

Consider semisimple linear algebraic groups over algebraically closed fields. These are precisely elementary Chevalley groups $E_{\pi }(\Phi ,K)$ (see. [59], Section 5).

All these groups are defined in $\text{SL}{ }_n(K)$ as common set of zeros of polynomials of matrix entries a_ij with integer coefficients (for example, in the case of the root system ${\mathbf{C}}_{\mathcal{l}}$ and the universal representation we have $n=2l$ and the polynomials from the condition $(a_{ij})Q(a_{ji})-Q=0$, where Q is a matrix of the symplectic form). It is clear now that multiplication and taking inverse element are defined by polynomials with integer coefficients. Therefore, these polynomials can be considered as polynomials over an arbitrary commutative ring with a unit. Let some elementary Chevalley group E over $\mathbb{C}$ be defined in $\text{SL}{ }_n(\mathbb{C})$ by polynomials $p_1(a_{ij}),\dots ,p_m(a_{ij})$. For a commutative ring R with a unit let us consider the group $$G(R)=\{(a_{ij})\in \text{SL}{ }_n(R)⏧{\tilde{p}}_1(a_{ij})=0,\dots ,{\tilde{p}}_m(a_{ij})=0\},$$where ${\tilde{p}}_1(\dots ),\dots {\tilde{p}}_m(\dots )$ are polynomials having the same coefficients as $p_1(\dots ),\dots ,p_m(\dots )$, but considered over R.

This group is called the Chevalley group $G_{\pi }(\Phi ,R)$ of the type $\Phi$ over the ring R, and for every algebraically closed field K it coincides with the elementary Chevalley group. In more advanced terms a Chevalley group $G(\Phi ,R)$ is the value of the Chevalley-Demazure group scheme, see [23].

The subgroup of diagonal (in the standard basis of weight vectors) matrices of the Chevalley group $G_{\pi }(\Phi ,R)$ is called the standard maximal torus of $G_{\pi }(\Phi ,R)$ and it is denoted by $T_{\pi }(\Phi ,R)$. This group is isomorphic to $\mathit{Hom}({\Lambda }_{\pi },R^{*})$.

Let us denote by $h(\chi )$ the elements of the torus $T_{\pi }(\Phi ,R)$, corresponding to the homomorphism $\chi \in \mathit{Hom}(\Lambda (\pi ),R^{*})$.

In particular, $h_{\alpha }(u)=h({\chi }_{\alpha ,u})$ ($u\in R^{*},\alpha \in \Phi$), where $${\chi }_{\alpha ,u}:\lambda \mapsto u^{〈\lambda ,\alpha 〉}\hspace{1em}(\lambda \in {\Lambda }_{\pi }).$$

2.4 Connection between Chevalley groups and their elementary subgroups

Connection between Chevalley groups and corresponding elementary subgroups is an important problem in the structure theory of Chevalley groups over rings. For elementary Chevalley groups there exists a convenient system of generators $x_{\alpha }(\xi ),\alpha \in \Phi ,\xi \in R$, and all relations between these generators are well-known. For general Chevalley groups it is not always true.

If R is an algebraically closed field, then $$G_{\pi }(\Phi ,R)=E_{\pi }(\Phi ,R)$$for any representation π. This equality is not true even for the case of fields, which are not algebraically closed.

However if G is a simply connected Chevalley group and the ring R is semilocal (i.e., contains only finite number of maximal ideals), then we have the condition $$G_{sc}(\Phi ,R)=E_{sc}(\Phi ,R).$$

[2, 4, 42, 58].

If, however, π is arbitrary and R is semilocal, then: $G_{\pi }(\Phi ,R)=E_{\pi }(\Phi ,R)T_{\pi }(\Phi ,R)$] (see [2, 4, 42]), and the elements $h(\chi )$ are connected with elementary generators by the formula (1) $$h(\chi )x_{\beta }(\xi )h{(\chi )}^{-1}=x_{\beta }(\chi (\beta )\xi ).$$(1)

Remark 1.

Since $\chi \in \text{Hom} (\Lambda (\pi ),R^{*})$, if we know the values of χ on some set of roots which generate all roots (for example, on some basis of $\Phi$), then we know $\chi (\beta )$ for all $\beta \in \Phi$ and respectively all $x_{\beta }{(\xi )}^{h(\chi )}$ for all $\beta \in \Phi$ and $\xi \in R^{*}$.

Therefore (in particular) if for all roots β from some generating set of $\Phi$ we have $[x_{\beta }(1),h(\chi )]=1$, then $h(\chi )\in Z(E_{\pi }(\Phi ,R))$ and hence $h(\chi )\in Z(G_{\pi }(\Phi ,R))$.

We will use this observation in the next section many times.

If $\Phi$ is an irreducible root system of a rank $\mathcal{l}⩾2$, then $E(\Phi ,R)$ is always normal and even characteristic in $G(\Phi ,R)$ (see [31, 63]). In the case of semilocal rings it is easy to show that $$[G(\Phi ,R),G(\Phi ,R)]=E(\Phi ,R).$$except the cases $\Phi ={\mathbf{B}}_2,{\mathbf{G}}_2,R={\mathbb{F}}_2$.

In the case $\mathcal{l}=1$ the subgroup of elementary matrices $E_2(R)=E_{sc}({\mathbf{A}}_1,R)$ is not necessarily normal in the special linear group $\text{SL}{ }_2(R)=G_{sc}({\mathbf{A}}_1,R)$ (see [24, 61, 62]).

In the general case the difference between $G_{\pi }(\Phi ,R)$ and $E_{\pi }(\Phi ,R)$ is measured by K₁-functor.

2.5 Standard automorphisms of Chevalley groups

Define four types of automorphisms of a Chevalley group $G_{\pi }(\Phi ,R)$, we call them standard.

Central automorphisms. Let $C_G(R)$ be a center of $G_{\pi }(\Phi ,R),\tau :G_{\pi }(\Phi ,R)\to C_G(R)$ be some homomorphism of groups. Then the mapping $x\mapsto \tau (x)x$ from $G_{\pi }(\Phi ,R)$ onto itself is an automorphism of $G_{\pi }(\Phi ,R)$, denoted by τ. It is called a central automorphism of the group $G_{\pi }(\Phi ,R)$.

Ring automorphisms. Let $\rho :R\to R$ be an automorphism of the ring R. The mapping $(a_{i,j})\mapsto (\rho (a_{i,j}))$ from $G_{\pi }(\Phi ,R)$ onto itself is an automorphism of the group $G_{\pi }(\Phi ,R)$, denoted by the same letter ρ. It is called a ring automorphism of the group $G_{\pi }(\Phi ,R)$. Note that for all $\alpha \in \Phi$ and $t\in R$ an element $x_{\alpha }(t)$ is mapped to $x_{\alpha }(\rho (t))$.

Inner automorphisms. Let S be some ring containing R, g be an element of $G_{\pi }(\Phi ,S)$, that normalizes the subgroup $G_{\pi }(\Phi ,R)$. Then the mapping $x\mapsto \mathit{gx}{g}^{-1}$ is an automorphism of the group $G_{\pi }(\Phi ,R)$, denoted by i_g. It is called an inner automorphism, induced by the element $g\in G_{\pi }(\Phi ,S)$. If $g\in G_{\pi }(\Phi ,R)$, then we call i_g a strictly inner automorphism.

Graph automorphisms. Let δ be an automorphism of the root system $\Phi$ such that $\delta \Delta =\Delta$. Then there exists a unique automorphisms of $G_{\pi }(\Phi ,R)$ (we denote it by the same letter δ) such that for every $\alpha \in \Phi$ and $t\in R$ an element $x_{\alpha }(t)$ is mapped to $x_{\delta (\alpha )}(\epsilon (\alpha )t)$, where $\epsilon (\alpha )=\pm 1$ for all $\alpha \in \Phi$ and $\epsilon (\alpha )=1$ for all $\alpha \in \Delta$.

Now suppose that ${\delta }_1,\dots ,{\delta }_k$ are all different graph automorphisms for the given root system (for the systems ${\mathbf{E}}_7,{\mathbf{E}}_8,{\mathbf{B}}_l,{\mathbf{C}}_l,{\mathbf{F}}_4,{\mathbf{G}}_2$ there can be just identical automorphism, for the systems ${\mathbf{A}}_l,{\mathbf{D}}_l,l\ne 4,{\mathbf{E}}_6$ there are two such automorphisms, for the system ${\mathbf{D}}_4$ there are six automorphisms). Suppose that we have a system of orthogonal idempotents of the ring R: $$\{{\epsilon }_1,\dots ,{\epsilon }_k⏧{\epsilon }_1+\dots +{\epsilon }_k=1,\forall i\ne j{\epsilon }_i{\epsilon }_j=0\}.$$

Then the mapping $${\Lambda }_{{\epsilon }_1,\dots ,{\epsilon }_k}:={\epsilon }_1{\delta }_1+\dots +{\epsilon }_k{\delta }_k$$of the Chevalley group onto itself is an automorphism, called a graph automorphism of the Chevalley group $G_{\pi }(\Phi ,R)$.

Similarly we can define four types of automorphisms of the elementary subgroup $E_{\pi }(\Phi ,R)$. An automorphism σ of the group $G_{\pi }(\Phi ,R)$ (or $E_{\pi }(\Phi ,R)$) is called standard if it is a composition of automorphisms of these introduced four types.

In [14] the following theorem was proved:

Theorem 1.

Let $G=G_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$ be an adjoint Chevalley group (or its elementary subgroup $(E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R))$) of rank > 1, R be a commutative ring with 1. Suppose that for $\Phi ={\mathbf{A}}_2,{\mathbf{B}}_l,{\mathbf{C}}_l$ or ${\mathbf{F}}_4$ we have $1/2\in R$, for $\Phi ={\mathbf{G}}_2$ we have $1/2,1/3\in R$. Then every automorphism of the group G is standard and the inner automorphism in the composition is strictly inner.

Our goal is to prove the following theorem:

Theorem 2.

Let $G=G_{\pi }(\Phi ,R)$ be a Chevalley group (or its elementary subgroup $E_{\pi }(\Phi ,R)))$ of rank > 1, R be a commutative ring with 1. Suppose that for $\Phi ={\mathbf{A}}_2,{\mathbf{B}}_l,{\mathbf{C}}_l$ or ${\mathbf{F}}_4$ we have $1/2\in R$, for $\Phi ={\mathbf{G}}_2$ we have $1/2,1/3\in R$. Then every automorphism of the group G is standard.

3 Proof of the main theorem for elementary Chevalley groups and subgroups

3.1 Localization of rings and modules; injection of a ring into the product of its localizations

Definition 1.

Let R be a commutative ring. A subset $Y\subset R$ is called multiplicatively closed in R, if $1\in Y$ and Y is closed under multiplication.

Introduce an equivalence relation ∼ on the set of pairs R × Y as follows: $$\frac{a}{s}\sim \frac{b}{t}\iff \exists u\in Y:(\mathit{at}-\mathit{bs})u=0.$$

By $\frac{a}{s}$ we denote the whole equivalence class of the pair (a, s), by $Y^{-1}R$ we denote the set of all equivalence classes. On the set $Y^{-1}R$ we can introduce the ring structure by $$\frac{a}{s}+\frac{b}{t}=\frac{at+bs}{st},\hspace{1em}\frac{a}{s}·\frac{b}{t}=\frac{ab}{st}.$$

Definition 2.

The ring $Y^{-1}R$ is called the ring of fractions of R with respect to Y.

Let $\mathfrak{p}$ be a prime ideal of R. Then the set $Y=R\setminus \mathfrak{p}$ is multiplicatively closed (it is equivalent to the definition of the prime ideal). We will denote the ring of fractions $Y^{-1}R$ in this case by $R_{\mathfrak{p}}$. The elements $\frac{a}{s},a\in \mathfrak{p}$, form an ideal $\mathfrak{M}$ in $R_{\mathfrak{p}}$. If $\frac{b}{t}\notin \mathfrak{M}$, then $b\in Y$, therefore $\frac{b}{t}$ is invertible in $R_{\mathfrak{p}}$. Consequently the ideal $\mathfrak{M}$ consists of all non-invertible elements of the ring $R_{\mathfrak{p}}$, i. e., $\mathfrak{M}$ is the greatest ideal of this ring, so $R_{\mathfrak{p}}$ is a local ring.

The process of passing from R to $R_{\mathfrak{p}}$ is called localization at $\mathfrak{p}$.

Proposition 1.

Every commutative ring R with 1 can be naturally embedded in the cartesian product of all its localizations by maximal ideals $$S=\prod _{\mathfrak{m}\text{ is a maximal ideal of }R}{R}_{\mathfrak{m}}$$by diagonal mapping, which corresponds every $a\in R$ to the element $\prod _{\mathfrak{m}}{(\frac{a}{1})}_{\mathfrak{m}}\in S.$

3.2 Proof for $E_{\pi }(\Phi ,R)$

Suppose that $G=G_{\pi }(\Phi ,R)$ or $G=E_{\pi }(\Phi ,R)$ is a Chevalley group (or its elementary subgroup), where $\Phi$ is an indecomposable root system of rank > 1, R is an arbitrary commutative ring (with 1/2 in the case $\Phi ={\mathbf{A}}_2,{\mathbf{F}}_4,{\mathbf{B}}_l,{\mathbf{C}}_l$ and with 1/2 and 1/3 in the case $\Phi ={\mathbf{G}}_2$). Suppose that $\phi \in \text{Aut} (G)$.

Since the subgroup $E_{\pi }(\Phi ,R)$ is characteristic in $G_{\pi }(\Phi ,R)$, then $\phi$ induces the automorphism $\phi \in \text{Aut} (E_{\pi }(\Phi ,R))$ (we denote it by the same letter).

The elementary adjoint Chevalley group $E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$ is the quotient group of our initial elementary Chevalley group $E_{\pi }(\Phi ,R)$ by its center $Z=Z(E_{\pi }(\Phi ,R))$. Therefore the automorphism $\phi$ induces an automorphism $\overline{\phi }$ of the adjoint Chevalley group $E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$. By Theorem 1 $\overline{\phi }$ is the composition of a graph automorphism ${\overline{\Lambda }}_{{\epsilon }_1,\dots ,{\epsilon }_k}$, where ${\epsilon }_1,\dots ,{\epsilon }_k\in R$, a ring automorphism $\overline{\rho }$, induced by $\rho \in \text{Aut} R$, and the strictly inner automorphism $i_{\overline{g}}$, induced by some $\overline{g}\in G_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$. Central automorphism is identical in the decomposition of $\overline{\phi }$, since the center of any adjoint Chevalley group is trivial.

Since ${\epsilon }_1,\dots ,{\epsilon }_k\in R$ and for any ${\delta }_i\in \text{Aut} \Delta$ and for any representation π of the corresponding Lie algebra there exists the corresponding graph automorphism ${\delta }_i\in \text{Aut} (G_{\pi }(\Phi ,R))$, then there exists a graph automorphism ${\Lambda }_{{\epsilon }_1,\dots ,{\epsilon }_k}\in \text{Aut} (E_{\pi }(\Phi ,R))$ such that the induced automorphism of the group $E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$ is precisely ${\overline{\Lambda }}_{{\epsilon }_1,\dots ,{\epsilon }_k}$.

Also taking the ring automorphism $\rho \in \text{Aut} (G_{\pi }(\Phi ,R))$ we see that the induced automorphism of $E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$ is precisely $\overline{\rho }$.

Therefore if we take ${\phi }_1={\Lambda }^{-1}°{\rho }^{-1}°\phi$, then we obtain an automorphism of the group G (and in any cases of the group/subgroup $E_{\pi }(\Phi ,R)$) which induces the strictly inner automorphism $i_{\overline{g}}$ on $E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$.

We always assume that R is a subring of the ring $S=\prod _{\mathfrak{m}}{R}_{\mathfrak{m}}=\prod _{i\in \varkappa }{R}_i$, where every R_i is a local ring, therefore $$G_{\pi }(\Phi ,R)\subseteq G_{\pi }(\Phi ,S)=\prod _{i\in \varkappa }{G}_{\pi }(\Phi ,R_i)\text{ and }{E}_{\pi }(\Phi ,R)\subseteq \prod _{i\in \varkappa }{E}_{\pi }(\Phi ,R_i).$$

Note that since every R_i is local, then we have $G_{\pi }(\Phi ,R)=T_{\pi }(\Phi ,R)E_{\pi }(\Phi ,R)$ and therefore $$\prod _{i\in \varkappa }{G}_{\pi }(\Phi ,R_i)=\prod _{i\in \varkappa }{T}_{\pi }(\Phi ,R_i)E_{\pi }(\Phi ,R_i).$$

Suppose now that $\overline{g}=\prod _{i\in \varkappa }{\overline{g}}_i$, where ${\overline{g}}_i\in G_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i)$.

Let us consider one $i\in \varkappa$, where ${\overline{g}}_i\in T_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i)E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i)$, i.e., $g_i={\overline{t}}_i·{\overline{x}}_i$, where ${\overline{t}}_i\in T_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i),{\overline{x}}_i\in E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i)$.

Since ${\overline{x}}_i$ is a product of elementary unipotents over the ring R_i, then we can take $x_i\in E_{\pi }(\Phi ,R_i)$, that is the same product of the same elementary unipotents and its image under factorization of $E_{\pi }(\Phi ,R_i)$ by its center is precisely ${\overline{x}}_i$.

Now let us consider the element ${\overline{t}}_i\in T_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i)$. This element corresponds to some homomorphism ${\chi }_i\in \text{Hom} (\Lambda ( \text{ad} ),R_i^{*})$ and acts on any $x_{\alpha }(s)\in E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i)$ as $${\overline{t}}_i{x}_{\alpha }(s){\overline{t}}_i^{-1}=x_{\alpha }({\chi }_i(\alpha )·s).$$

If ${\overline{t}}_i\notin H_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R_i)$, then we can extend the ring R_i up to a ring S_i so that there exists ${\overline{h}}_i\in H_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,S_i)$ with the same action on all elementary uniponents $x_{\alpha }(s)$ as our ${\overline{t}}_i$. The ring S_i is an algebraic extension of R_i, in which there exist several new roots $\sqrt[k]{\lambda }$ for a finite number of $\lambda \in R_i^{*}$. This S_i can be obtained from R_i by the standard procedure $$S_i\cong R_i\left[y\right]/(y^k-\lambda ).$$

Note that S_i is not necessarily local.

Now since $R_i\subseteq S_i$, then $S\subseteq \prod _{i\in \varkappa }{S}_i=\tilde{S}$ and $R\subseteq S\subseteq \tilde{S}$. We see that for every $i\in \varkappa$ the torus element ${\overline{t}}_i$ acts on all $x_{\alpha }(s),s\in S_i$ as ${\overline{h}}_i\in H_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,S_i)$, therefore the element ${\overline{y}}_i={\overline{h}}_i·{\overline{x}}_i$ acts on all $x_{\alpha }(s),s\in S_i$ as the initial ${\overline{g}}_i$.

Consequently the element $\overline{y}:=\prod _{i\in \varkappa }{\overline{y}}_i\in E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,\tilde{S})$ acts on all $x_{\alpha }(s),s\in \tilde{S}$ as the initial $\overline{g}$.

Therefore we have $\overline{y}\in E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,\tilde{S})$ such that $$i_{\overline{y}}{|}_{E_{ \text{ad} }(\Phi ,\tilde{S})}=i_{\overline{g}}{|}_{E_{ \text{ad} }(\Phi ,\tilde{S})}.$$

In particular, $$i_{\overline{y}}{|}_{E_{ \text{ad} }(\Phi ,R)}=i_{\overline{g}}{|}_{E_{ \text{ad} }(\Phi ,R)}.$$

Let us take $y\in E_{\pi }(\Phi ,\tilde{S})$ such that its image under factorization of $E_{\pi }(\Phi ,\tilde{S})$ by its center is precisely $\overline{y}$.

Now we can take ${\phi }_2=i_{y^{-1}}°{\phi }_1$, it will be an isomorphism between $E_{\pi }(\Phi ,R)$ and the subgroup of $E_{\pi }(\Phi ,\tilde{S})$ such that under factorization by the center of $E_{\pi }(\Phi ,\tilde{S})$ we obtain the identical automorphism ${\overline{\phi }}_2$ of the group $E_{\mathrm{ }\text{ad}\mathrm{ }}(\Phi ,R)$.

Now let us analyze the mapping ${\phi }_2$.

Since ${\overline{\phi }}_2$ is identical, then $$\forall \alpha \in \Phi \forall s\in \mathit{R\hspace{1em}}{\phi }_2(x_{\alpha }(s))=z_{\alpha ,s}{x}_{\alpha }(s),\text{ where }{z}_{\alpha ,s}\in Z(E_{\pi }(\Phi ,\tilde{S})).$$

If α is either any root of the systems ${\mathbf{A}}_l$, $\mathit{l⩾}2,{\mathbf{D}}_l,\mathit{l⩾}4,{\mathbf{E}}_l$, l = 6, 7, 8, ${\mathbf{F}}_4$, or any long root of the systems ${\mathbf{G}}_2,{\mathbf{B}}_l,\mathit{l⩾}3$, or any short root of the systems ${\mathbf{C}}_l,\mathit{l⩾}3$, then α can be represented as $\alpha =\beta +\gamma$, where $\{\pm \beta ,\pm \gamma ,\pm \alpha \}\cong {\mathbf{A}}_2$. In this case $$x_{\alpha }(s)=[x_{\beta }(s),x_{\gamma }(1)],$$

therefore $$z_{\alpha ,s}{x}_{\alpha }(s)={\phi }_2(x_{\alpha }(s))=[{\phi }_2(x_{\beta }(s)),{\phi }_2(x_{\gamma }(1))]=[z_{\beta ,s}{x}_{\beta }(s),z_{\gamma ,1}{x}_{\gamma }(1)]=[x_{\beta }(s),x_{\gamma }(1)]=x_{\alpha }(s).$$

Consequently, $z_{\alpha ,s}=1$ for all $s\in R$.

For the root system ${\mathbf{G}}_2$ all Chevalley groups are adjoint and so we do not need to prove Theorem 1 for this root system.

For the root system ${\mathbf{B}}_2$ if α is a long simple root and β is a short simple root, then ${\Phi }^{+}=\{\alpha ,\beta ,\alpha +\beta ,\alpha +2\beta \}$, where $\alpha +\beta$ is short and $\alpha +2\beta$ is long and $$\begin{array}{c}[x_{\alpha }(t),x_{\beta }(u)]=x_{\alpha +\beta }(\pm \mathit{tu})x_{\alpha +2\beta }(\pm t{u}^2),\\ [x_{\alpha +\beta }(t),x_{\beta }(u)]=x_{\alpha +2\beta }(\pm 2\mathit{tu}).\end{array}$$

(see [59], Lemma 33).

Since for the root system ${\mathbf{B}}_2$ we require $1/2\in R$, then $$[x_{\alpha +\beta }(s),x_{\beta }(1/2)]=x_{\alpha +2\beta }(\pm s)$$

and by the same arguments as above $z_{\gamma ,s}=1$ for all long roots γ and all $s\in R$. Then $$\begin{array}{c}{x}_{\alpha +\beta }(\pm s)x_{\alpha +2\beta }(\pm s)=[x_{\alpha }(s),x_{\beta }(1)]={\phi }_2([x_{\alpha }(s),x_{\beta }(1)])=\\ ={\phi }_2(x_{\alpha +\beta }(\pm s)x_{\alpha +2\beta }(\pm s))=z_{\alpha +\beta ,\pm s}{x}_{\alpha +\beta }(\pm s)x_{\alpha +2\beta }(\pm s),\end{array}$$thus $z_{\gamma ,s}=1$ also for all short roots $\gamma \in {\mathbf{B}}_2$. Therefore for ${\mathbf{B}}_2$ for all $\alpha \in \Phi$ and all $s\in R$ the mapping ${\phi }_2$ is an identical automorphism of $E_{\pi }(\Phi ,R)$.

Since any root γ of the root system ${\mathbf{B}}_l$ or ${\mathbf{C}}_l,\mathit{l⩾}3$, can be embedded to some root system isomorphic to ${\mathbf{B}}_2$, and in this case we also require $1/2\in R$, then for these root systems also $z_{\gamma ,s}=1$ for all $s\in R^{*}$ and ${\phi }_2$ is an identical automorphism of $E_{\pi }(\Phi ,R)$.

Therefore for all cases under consideration $${\phi }_2{|}_{E_{\pi }(\Phi ,R)}=i_{y^{-1}}°{\Lambda }^{-1}°{\rho }^{-1}°\phi {|}_{E_{\pi }(\Phi ,R)}=i{d}_{E_{\pi }(\varphi ,R)},$$

so $$\phi {|}_{E_{\pi }(\Phi ,R)}=\rho °\Lambda °i_y{|}_{E_{\pi }(\Phi ,R)},$$where $y\in E_{\pi }(\Phi ,\tilde{S})\cap N(E_{\pi }(\Phi ,R))$, Λ is a graph automorphism of the groups $G_{\pi }(\Phi ,R)$ and $E_{\pi }(\Phi ,R)$ and ρ is a ring automorphism of the groups $G_{\pi }(\Phi ,R)$ and $E_{\pi }(\Phi ,R)$.

Thus, for $G=E_{\pi }(\Phi ,R)$ the main theorem (Theorem 2) is proved.

4 Proof of the main theorem for the groups $G_{\pi }(\phi ,R)$

Let now $G=G_{\pi }(\Phi ,R)$. Initially the mapping $\phi$ was an automorphism of the group G. The mapping ${\phi }_1$ from the previous section was the composition of $\phi$ and graph and ring automorphisms of the group G, i.e., also an automorphism of G. After that ${\phi }_2$ (from the previous section) is the composition of ${\phi }_1$ and the conjugation of G by some element $y\in E_{\pi }(\Phi ,\tilde{S})$, where $R\subset \tilde{S}$. We know that y normalizes $E_{\pi }(\Phi ,R)$ and we want to show that in our case y normalizes also our full Chevalley group G.

Note that for the simply-connected Chevalley group of the type ${\mathbf{E}}_6$ Luzgarev and Vavilov in [40] proved that the normalizers of the Chevalley group and its elementary subgroup coincide. Then in [41] they proved the same theorem for the root system ${\mathbf{E}}_7$. Since all other exceptional Chevalley groups are adjoint, we only need to show the coincidence of normalizers for non-adjoint classical Chevalley groups, but our method will cover all the cases.

Lemma 1.

Under assumptions of Theorem 2 the elements $x_{\alpha }(1),\alpha \in \Phi$, by addition, multiplication and multiplication by elements from R generate the Lie algebra $\pi ({\mathcal{L}}_R(\Phi ))\subset M_N(R)$, where N is the dimension of the representation π.

Proof.

For the adjoint Chevalley groups this lemma was proved in [14]. Therefore we will not repeat the proof for the root system ${\mathbf{G}}_2$ (since it is always adjoint).

If the root system differs from ${\mathbf{G}}_2$ and $1/2\in R$, then $x_{\alpha }(1)=E+\pi (X_{\alpha })+\pi {(X_{\alpha })}^2/2$, therefore $$\pi (X_{\alpha })=x_{\alpha }(1)-E-{(x_{\alpha }(1)-E)}^2/2,$$and $$\pi ({\mathcal{L}}_R(\Phi ))={〈\pi (X_{\alpha })\mathit{⏧\alpha }\in \Phi 〉}_R.$$

Suppose now that we deal with systems ${\mathbf{A}}_l$, ($\mathit{l⩾}3$), ${\mathbf{D}}_l,{\mathbf{E}}_l,1/2\notin R$.

For all these systems and non-adjoint representations π we have $\pi {(X_{\alpha })}^2=0$ for all $\alpha \in \Phi$, therefore $$\pi (X_{\alpha })=x_{\alpha }(1)-E.$$

The lemma is proved. □

From Lemma 1 we see that the conjugation by y maps the Lie algebra $\pi (\mathcal{L}{(\Phi )}_R)$ onto itself.

Lemma 2.

Under assumptions of Theorem 2 the Lie algebra $\pi ({\mathcal{L}}_R(\Phi ))$ together with the unity matrix E by addition, multiplication and multiplication by elements from R generate the matrix ring $M_N(R)$, where N is the dimension of the representation π.

Proof.

For all adjoint Lie algebras under consideration this fact was proved in the papers [8, 9, 11–13].

For classical representations of classical Lie algebras the proof is clear and direct:

1. If we have the root system ${\mathbf{A}}_l$ and the standard representation, then

   $\pi (X_{e_i-e_j})=E_{ij},\hspace{1em}\pi (X_{e_i-e_j})\pi (X_{e_j-e_i})=E_{ii},\hspace{1em}{M}_{l+1}(R)={〈E_{ij}⏧1\mathit{⩽i},\mathit{j⩽}l+1〉}_R.$

2. The Lie algebra of the type ${\mathbf{C}}_l$ in its universal representation has 2l-dimensional linear space and the basis

   $\{E_{ii}-E_{l+i,l+i};E_{ij}-E_{l+j,l+i};E_{i,l+i};E_{l+i,i};E_{i,l+j}+E_{j,l+i};E_{l+i,j}+E_{l+j,i}⏧1\mathit{⩽i}\ne \mathit{j⩽}l\}.$

   Multiplying $E_{ij}-E_{l+j,l+i}$ by $E_{j,l+j}$, we get all $E_{i,l+j}$ for all $1\mathit{⩽i},\mathit{j⩽}l$. Multiplying $E_{l+i,i}$ by $E_{ij}-E_{l+j,l+i}$, we obtain $E_{l+i,j}$ for all $1\mathit{⩽i},\mathit{j⩽}l$. It is clear that after that we have all E_ij, $1\mathit{⩽i},\mathit{j⩽}l$, and therefore the whole matrix ring $M_{2l}(R)$.

3. For the root system ${\mathbf{D}}_l$ the standard representation gives the algebra $\mathfrak{s}{\mathfrak{o}}_{2l}$, where in 2l-dimensional space the basis is

   $\{E_{ii}-E_{l+i,l+i};E_{ij}-E_{l+j,l+i};E_{i,l+j}-E_{j,l+i};E_{i+l,j}-E_{j+l,i}⏧1\mathit{⩽i}\ne \mathit{j⩽}l\}.$

Since for $i\ne j$ we have $(E_{ii}-E_{l+i,l+i})·(E_{ij}-E_{l+j,l+i})=E_{ij}$, then the whole matrix ring $M_{2l}(R)$ is generated by this Lie algebra.

All other representations are described by Plotkin, Semenov and Vavilov in [51] as microweight representations with the help of so-called weight diagrams.

Weight diagram is a labeled graph, its vertices correspond (bijectively) to the weights $\lambda \in \Lambda (\pi )$. The vertices corresponding to $\lambda ,\mu \in \Lambda (\pi )$, are joined by a bond marked ${\alpha }_i\in \Delta$ (or simply i) if and only if their difference $\lambda -\mu ={\alpha }_i$ is a simple root. The diagrams are usually drawn in such way that the marks on the opposite (parallel) sides of a parallelogram are equal and at least one of them is usually omitted. All weights are numbered in any order and give the basis of our representation π. If we want to find $\pi (X_{{\alpha }_i}),i=1,\dots ,l$, then we need to find all bonds marked by i, and if they join the vertices $({\gamma }_1,{\gamma }_1+{\alpha }_i),\dots ,({\gamma }_k,{\gamma }_k+{\alpha }_i)$, then $$\pi (X_{{\alpha }_i})=\pm E_{{\gamma }_1,{\gamma }_1+{\alpha }_i}\pm \cdots \pm E_{{\gamma }_k,{\gamma }_k+{\alpha }_i},\hspace{1em}\pi (X_{-{\alpha }_i})=\pm E_{{\gamma }_1+{\alpha }_i,{\gamma }_1}\pm \cdots \pm E_{{\gamma }_k+{\alpha }_i,{\gamma }_k}.$$

It is clear that if we take an element $\pi (X_{{\alpha }_i})·\pi (X_{{\alpha }_j})$, then it is a sum of $\pm E_{\gamma ,\gamma \prime }$, where there exists a path from the weight γ to $\gamma \prime$ of the length 2 marked by the sequence (i, j). Similarly, if we take an element $\pi (X_{{\alpha }_{i_1}})\times \cdots \times \pi (X_{{\alpha }_{i_k}})$, then it is a sum of $\pm E_{\gamma ,\gamma \prime }$, where there exists a path from the weight γ to $\gamma \prime$ of the length k marked by the sequence $(i_1,\dots ,i_k)$.

Our goal is to generate all matrix units $E_{{\gamma }_1,{\gamma }_2}$, where ${\gamma }_1,{\gamma }_2\in \Lambda (\pi )$. Since all weight diagrams are connected, it is sufficient to generate all matrix units $E_{\gamma ,\gamma +{\alpha }_i}$ and $E_{\gamma +{\alpha }_i,\gamma }$, where ${\alpha }_i\in \Delta ,\gamma ,\gamma +{\alpha }_i\in \Lambda (\pi )$. The general idea how to do it is the following: for any $\gamma \in \Lambda (\pi )$ and any ${\alpha }_{i_0}\in \Delta$ such that $\gamma +{\alpha }_{i_0}\in \Lambda (\pi )$ we find $\gamma \prime \in \Lambda (\pi )$ such that:

1. there exists a path $(i_0,i_1,\dots ,i_k)$ from γ to $\gamma \prime$;

2. in our weight diagram there is no other path $(i_0,i_1,\dots ,i_k)$;

3. the path $(i_1,\dots ,i_k)$ exists only from $\gamma +{\alpha }_i$ to $\gamma \prime$.

Then $$\pi (X_{{\alpha }_{i_0}})\pi (X_{{\alpha }_{i_1}})\dots \pi (X_{{\alpha }_{i_k}})=\pm E_{\gamma ,\gamma \prime }$$

and $$\pi (X_{-{\alpha }_{i_k}})\dots \pi (X_{-{\alpha }_{i_1}})=\pm E_{\gamma \prime ,\gamma +{\alpha }_{i_0}}$$and therefore $E_{\gamma ,\gamma +{\alpha }_{i_0}}=E_{\gamma ,\gamma \prime }{E}_{\gamma \prime ,\gamma +{\alpha }_{i_0}}$ can be generated.

It is almost clear that such $\gamma \prime$ and unique paths always exist, we will just show one diagram as an example.

If we take the case ${\mathbf{A}}_7$ with the weight ω₂, the representation is 28-dimensional. Let us find a path which gives $E_{{\gamma }_1,{\gamma }_2}$. Since the path (1, 3) is unique in the diagram, then the path (2, 1, 3) is also unique and we have $$E_{{\gamma }_1,{\gamma }_2}=(\pi (X_{{\alpha }_2})\pi (X_{{\alpha }_1})\pi (X_{{\alpha }_3}))·(\pi (X_{-{\alpha }_3})\pi (X_{-{\alpha }_1}).$$

If we want to generate, for example, $E_{{\gamma }_4,{\gamma }_6}$, then the suitable path is (4, 1, 5), since the path (1, 5) is unique in the diagram.

Looking at the picture it is easy to find the suitable path for any pair of neighboring vertices.

Fig. 1 ${\mathbf{A}}_7$, ω₂

Therefore the lemma is proved for all the cases. □

Since $\mathit{y\pi }(\mathcal{L}{(\Phi )}_R)y^{-1}=\pi (\mathcal{L}{(\Phi )}_R$ and $\pi (\mathcal{L}{(\Phi )}_R$ generates the whole matrix ring $M_N(R)$, then $y{M}_N(R)y^{-1}=M_N(R)$. Therefore $y{G}_{\pi }(\Phi ,R)y^{-1}\subseteq \text{SL}{ }_N(R)$. From the other side, since $y\in G_{\pi }(\Phi ,\tilde{S})$, then $y{G}_{\pi }(\Phi ,R)y^{-1}\subseteq G_{\pi }(\Phi ,\tilde{S})$. Since $G_{\pi }(\Phi ,\tilde{S})\cap \text{SL}{ }_N(R)$ is (by definition) the Chevalley group $G_{\pi }(\Phi ,R)$, then y normalizes G.

Now we know that ${\phi }_2$ is an automorphism of $G=G_{\pi }(\Phi ,R)$, identical on the elementary subgroup $E=E_{\pi }(\Phi ,R)$. Let us take some $g\in G$ and $x_1\in E$ and let $g{x}_1{g}^{-1}=x_2\in E$. Then $${\phi }_2(g){\phi }_2(x_1){\phi }_2{(g)}^{-1}={\phi }_2(x_2)\Rightarrow {\phi }_2(g)x_1{\phi }_2{(g)}^{-1}=x_2,$$therefore $${\phi }_2(g)x_1{\phi }_2{(g)}^{-1}=g{x}_1{g}^{-1}\Rightarrow (g^{-1}{\phi }_2(g))x_1{(g^{-1}{\phi }_2(g))}^{-1}=x_1,$$ so $$g^{-1}{\phi }_2(g)\in C_G(E).$$

By the main theorem from [3] $C_G(E)=Z(G)$, therefore $${\phi }_2(g)=c_g·g,\hspace{1em}{c}_g\in Z(G)\text{ for all }g\in G.$$

Whence ${\phi }_2$ is a central automorphism of G and the initial $\phi$ is the composition of graph, ring, inner and central automorphisms, i.e., $\phi$ is standard.

The theorem is proved.

5 Some applications: isomorphisms and model theory of Chevalley groups

Standard description of automorphisms of Chevalley groups allows to describe and classify Chevalley groups up to different type of equivalencies and also to study model-theoretic properties.

Theorem 3.

Let $G_1=G_{{\pi }_1}({\Phi }_1,R_1)$ and $G_2=G_{{\pi }_2}({\Phi }_2,R_2)$ be two Chevalley groups of ranks > 1, R₁, R₂ be commutative rings with 1. Suppose that for ${\Phi }_1={\mathbf{A}}_2,{\mathbf{B}}_l,{\mathbf{C}}_l$ or ${\mathbf{F}}_4$ we have $1/2\in R_1$, for ${\Phi }_1={\mathbf{G}}_2$ we have $1/2,1/3\in R_1$. Then every isomorphism between the groups G₁ and G₂ is standard: it is a composition of inner, diagram and central automorphisms of G₁ and ring isomorphism between G₁ and G₂.

Proof.

We will use Theorem 6 from [15]:

If $G_{{\pi }_1}({\Phi }_1,R_1)$ and $G_{{\pi }_2}({\Phi }_2,R_2)$ are Chevalley groups of ranks > 1, $\phi :G_{{\pi }_1}({\Phi }_1,R_1)\to G_{{\pi }_2}({\Phi }_2,R_2)$ is an isomorphism between them, then $\phi (E_{{\pi }_1}({\Phi }_1,R_1))=E_{{\pi }_2}({\Phi }_2,R_2)$.

Now together with an isomorphism $G_1\to G_2$ we have an isomorphism $\phi :E_{{\pi }_1}({\Phi }_1,R_1)\to E_{{\pi }_2}({\Phi }_2,R_2)$ between their elementary subgroups. Since the quotient groups of $E_{{\pi }_1}({\Phi }_1,R_1)$ and $E_{{\pi }_2}({\Phi }_2,R_2)$ by their centers are adjoint elementary Chevalley groups $E_{\mathrm{ }\text{ad}\mathrm{ }}({\Phi }_1,R_1)$ and $E_{\mathrm{ }\text{ad}\mathrm{ }}({\Phi }_2,R_2)$, then $\phi$ induces an isomorphism $\overline{\phi }:E_{\mathrm{ }\text{ad}\mathrm{ }}({\Phi }_1,R_1)\to E_{\mathrm{ }\text{ad}\mathrm{ }}({\Phi }_2,R_2)$. By Theorem 9 from [15] in this case ${\Phi }_1\cong {\Phi }_2,R_1\cong R_2$ and the isomorphism $\overline{\phi }$ is a composition of a ring isomorphism $\rho :E_{\mathrm{ }\text{ad}\mathrm{ }}({\Phi }_1,R_1)\to E_{\mathrm{ }\text{ad}\mathrm{ }}({\Phi }_1,R_2)$ induced by some isomorphism of rings $\rho :R_1\to R_2$ and an automorphism $\psi \in \text{Aut} (E_{\mathrm{ }\text{ad}\mathrm{ }}({\Phi }_1,R_2))$.

The ring isomorphism ρ also induces an isomorphism $\tilde{\rho }$ between the initial Chevalley groups G₁ and G₂. Therefore $\tilde{\phi }:={\tilde{\rho }}^{-1}°\phi$ is an automorphism of the Chevalley group G₁, which is by Theorem 2 a composition of ring, inner, diagram and central automorphisms of G₁, therefore the initial $\phi$ is a composition of inner, diagram and central automorphisms of G₁ and ring isomorphism between G₁ and G₂. □

Remark 2.

The result of Theorem 3 is valid with respect to elementary Chevalley groups $E_{{\pi }_1}({\Phi }_1,R_1)$ and $E_{{\pi }_2}({\Phi }_2,R_2)$ as well.

Corollary 1

(classification of Chevalley groups up to isomorphism). Under conditions from Theorem 3 two Chevalley groups G₁ and G₂ (elementary Chevalley groups, respectively) are isomorphic if and only if they have the same root systems ${\Phi }_1$ and ${\Phi }_2$, same weight lattices ${\Lambda }_{{\pi }_1}$ and ${\Lambda }_{{\pi }_2}$ and isomorphic rings R₁ and R₂.

Proof.

If $G_1\cong G_2$, then there exists an isomorphism $\phi :G_1\to G_2$, which is composition of a ring isomorphism $\rho :G_1\to G_2$ and some automorphism $\psi \in \text{Aut} G_1$ (according to Theorem 3). Therefore there exists a ring isomorphism between G₁ and G₂, i.e., G₁ and G₂ have the same root systems, weight lattices and isomorphic rings. □

Another application of Theorem 3 is classification of Chevalley groups up to elementary equivalence (for adjoint Chevalley groups it was done in [15]).

Definition 3.

Two algebraic systems ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ of the same language $\mathcal{L}$ are called elementarily equivalent, if their first order theories coincide.

Theorem 4

(Keisler–Shelah Isomorphism theorem, [36, 56]). Two models ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ of the same language are elementarily equivalent if and only if there exists an ultrafilter $\mathcal{F}$ such that $$\prod _{\mathcal{F}}{\mathcal{M}}_1\cong \prod _{\mathcal{F}}{\mathcal{M}}_2.$$

Corollary 2 (classification of Chevalley groups up to elementary equivalence). Under conditions from Theorem 3 two Chevalley groups G₁ and G₂ (elementary Chevalley groups, respectively) are elementarily equivalent if and only if they have the same root systems ${\Phi }_1$ and ${\Phi }_2$, same weight lattices ${\Lambda }_{{\pi }_1}$ and ${\Lambda }_{{\pi }_2}$ and elementarily equivalent rings R₁ and R₂.

Proof.

By Theorem 4 the groups G₁ and G₂ are elementarily equivalent if and only if for some ultrafilter $\mathcal{F}$ their ultrapowers are isomorphic. Since $$\prod _{\mathcal{F}}{G}_{\pi }(\Phi ,R)\cong G_{\pi }(\Phi ,\prod _{\mathcal{F}}R),$$the latter is equivalent to $$G_{{\pi }_1}({\Phi }_1,\prod _{\mathcal{F}}{R}_1)\cong G_{{\pi }_2}({\Phi }_2,\prod _{\mathcal{F}}{R}_2)\iff \{\begin{array}{c}{\Lambda }_{{\pi }_1}={\Lambda }_{{\pi }_2},\hfill \\ {\Phi }_1={\Phi }_2,\hfill \\ \prod _{\mathcal{F}}{R}_1\cong \prod _{\mathcal{F}}{R}_2,\hfill \end{array}\iff \{\begin{array}{c}{\Lambda }_{{\pi }_1}={\Lambda }_{{\pi }_2},\hfill \\ {\Phi }_1={\Phi }_2,\hfill \\ R_1\equiv R_2,\hfill \end{array}$$ what was required. □

Two last corollaries almost finalize classification of Chevalley groups over commutative rings up to isomorphisms and elementary equivalence. However, there are still open questions concerning the relations of Chevalley groups with model theory.

In the recent work of D. Segal and K. Tent [55] the question of bi-interpretability of Chevalley groups over integral domains was considered (see [55] and [37] for the definition of bi-interpretability):

Theorem 5.

[55] Let $G(R)=G_{\pi }(\Phi ,R)$ be a Chevalley group of rank atleast two, and let R be an integral domain. Then R and G(R) are bi-interpretableprovided either

1. G is adjoint, or

2. G(R) has finite elementary width,

assuming in case $\Phi ={\mathbf{E}}_6,{\mathbf{E}}_7,{\mathbf{E}}_8$, or ${\mathbf{F}}_4$ that R has at least two units.

In the paper [16] regular bi-interpretabilty of Chevalley groups over local rings was obtained. This result used the ideas from [55] along with description of isomorphisms between Chevalley groups over local rings. It has also been proved that the class of Chevalley groups over local rings is elementarily definable: any group that is elementarily equivalent to some Chevalley group over a local ring is also a Chevalley group (of the same type) over a local ring (see [16]). Theorems 2 and 3 of the current paper allows us to prove regular bi-interptretability and elementary definability of adjoint Chevalley groups and Chevalley groups of finite elementary width over arbitrary commutative rings.

Acknowledgments

My sincere thanks go to Eugene Plotkin for very useful discussions regarding various aspects of this work and permanent attention to it.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.