Classification of certain types of maximal matrix subalgebras

TL;DR

This paper classifies maximal nonunital intersection subalgebras of matrix algebras over a field of characteristic zero, establishing their maximum dimension and providing a complete classification for those attaining it.

Contribution

It introduces a complete classification of maximal nonunital intersection subalgebras in matrix algebras, extending recent work and identifying their maximum dimension.

Findings

Maximum dimension of nonunital intersections is (n-1)(n-2) for n ≥ 3.

Complete classification of nonunital intersections of maximum dimension.

Classification of maximal unital subalgebras within parabolic subalgebras.

Abstract

Let $M_n(K)$ denote the algebra of $n \times n$ matrices over a field $K$ of characteristic zero. A nonunital subalgebra $N \subset M_n(K)$ will be called a nonunital intersection if $N$ is the intersection of two unital subalgebras of $M_n(K)$. Appealing to recent work of Agore, we show that for $n \ge 3$, the dimension (over $K$) of a nonunital intersection is at most $(n-1)(n-2)$, and we completely classify the nonunital intersections of maximum dimension $(n-1)(n-2)$. We also classify the unital subalgebras of maximum dimension properly contained in a parabolic subalgebra of maximum dimension in $M_n(K)$.

Equations

$$AM{A}^{-1}\in \mathcal{M}[R_n,C_n]\cap A\mathcal{M}[R_n,C_n]A^{-1}\text{mbox}ifandonlyifAM{A}^{-1}\in \mathcal{W}.$$

$$\mathcal{P}={\mathcal{P}}_K=\mathcal{M}[R_n]+K{E}_{n,n},{\mathcal{P}}^{\prime }={\mathcal{P}}_K^{\prime }=\mathcal{M}[C_1]+K{E}_{1,1}.$$

$$\mathcal{M}[R_n]=\sum _{a=1}^{n-1}\sum _{b=1}^{n}K{E}_{a,b}\subset \mathcal{L},$$

$$B:=\sum _{i=1}^n{c}_i{E}_{n,i},c_i\in K.$$

$$\mathcal{U}=e\mathcal{U}e\subset e\mathcal{M}e=D_r\mathcal{M}{D}_r\subset D_{n-1}\mathcal{M}{D}_{n-1}=\mathcal{M}[R_n,C_n].$$

$$\mathcal{M}{E}_{n,n}+\mathcal{M}[R_n,C_1]+K{E}_{1,1},\mathcal{M}{E}_{n,n}+\mathcal{M}[R_n,R_{n-1}]+K{E}_{n-1,n-1}.$$

$$\mathcal{C}=\mathcal{B}+K{E}_{n,n},\mathcal{D}=\mathcal{M}[R_n]E_{n,n}.$$

$$\dim \mathcal{B}\le \dim \mathcal{C}=\dim \mathcal{P}-\dim \mathcal{D}=(n^2-n+1)-(n-1)=n^2-2n+2.$$

$$\dim \mathcal{B}<n^2-2n+3,$$

$$d:=\dim \mathcal{B}e\le (n-1)(n-2)+1,$$

$$\dim \mathcal{L}\le (n-1)(n-2)+1$$

$$\dim \mathcal{L}\le (n-1)(n-2)<(n-1)(n-2)+1$$

$$\dim \mathcal{L}\le \dim \mathcal{M}[R_{n-1},C_{n-1}]=(n-2{)}^2<(n-1)(n-2)+1.$$

$$\dim \mathcal{L}\le (n-1)(n-2)+1.$$

$$B_1,\dots ,B_d,E_{1,n},\dots ,E_{n,n},$$

$$\dim \mathcal{B}\le d+n\le (n-1)(n-2)+1+n=n^2-2n+3.$$

$$\dim \mathcal{B}=d+n=(n-1)(n-2)+1+n=n^2-2n+3,$$

$$\mathcal{B}=\mathcal{B}e+\mathcal{M}{E}_{n,n}.$$

$$\mathcal{M}[R_n,C_n,C_1]+K{E}_{1,1},\mathcal{M}[R_n,C_n,R_{n-1}]+K{E}_{n-1,n-1}.$$

$$\left\{X_1(z),...,X_{n-1}(z)\right\}$$

$${\mu }_1(z)\cdots {\mu }_{n-1}(z)y_1\wedge \cdots \wedge y_{n-1}\ne 0\text{. }$$

$$h[R{\otimes }_F\overline{F},R{\otimes }_F\overline{F}]h^{-1}$$

Full text

2010 Mathematics Subject Classification. 15B33, 16S50

Key words and phrases. matrix ring over a field, intersection of matrix subalgebras, nonunital intersections, subalgebras of maximum dimension, parabolic subalgebra, semi-simple Lie algebra, radical

1 Introduction

Let $\mathcal{M}_{n}(F)$ denote the algebra of $n\times n$ matrices over a field $F$. For some interesting sets $\Lambda$ of subspaces $\mathcal{S}\subset\mathcal{M}_{n}(F)$, those $\mathcal{S}\in\Lambda$ of maximum dimension over $F$ have been completely classified. For example, a theorem of Gerstenhaber and Serezhkin [6, Theorem 1] states that when $\Lambda$ is the set of subspaces $\mathcal{S}\subset\mathcal{M}_{n}(F)$ for which every matrix in $\mathcal{S}$ is nilpotent, then each $\mathcal{S}\in\Lambda$ of maximum dimension is conjugate to the algebra of all strictly upper triangular matrices in $\mathcal{M}_{n}(F)$. For another example, it is shown in [1, Prop. 2.5] that when $\Lambda$ is the set of proper unital subalgebras $\mathcal{S}\subset\mathcal{M}_{n}(F)$ and $F$ is an algebraically closed field of characteristic zero, then each $\mathcal{S}\in\Lambda$ of maximum dimension is a parabolic subalgebra of maximum dimension in $\mathcal{M}_{n}(F)$.

The goal of this paper is to classify the elements in $\Lambda$ of maximum dimension in the cases $\Lambda=\Gamma$ and $\Lambda=\Omega$, where the sets $\Gamma$ and $\Omega$ are defined below.

In Isaac’s text [3, p. 161], every ring is required to have a unity, but the unity in a subring need not be the same as the unity in its parent ring. Under this definition, a ring may have subrings whose intersection is not a subring. This motivated us to study examples of pairs of unital subrings in $\mathcal{M}_{n}(K)$ whose intersection $\mathcal{N}$ is nonunital, where $K$ is a field of characteristic zero. We call such $\mathcal{N}$ a nonunital intersection and we let $\Gamma$ denote the set of all nonunital intersections $\mathcal{N}\subset\mathcal{M}_{n}(K)$. Note that $\Gamma$ is closed under transposition and conjugation, i.e., if $\mathcal{N}\in\Gamma$, then $\mathcal{N}^{\,\rm{T}}\in\Gamma$ and $S^{-1}\mathcal{N}S\in\Gamma$ for any invertible $S\in\mathcal{M}_{n}(K)$.

In order to define $\Omega$, we need to establish some notation. For brevity, write $\mathcal{M}=\mathcal{M}_{n}=\mathcal{M}_{n}(K)$. In the spirit of [2, p. viii], we define a subalgebra of $\mathcal{M}$ to be a vector subspace of $\mathcal{M}$ over $K$ closed under the multiplication of $\mathcal{M}$ (cf. [2, p. 2]); thus a subalgebra need not have a unity, and the unity of a unital subalgebra need not be a unity of the parent algebra. Subalgebras $\mathcal{A},\ \mathcal{B}\subset\mathcal{M}$ are said to be similar if $\mathcal{A}=\{S^{-1}BS:B\in\mathcal{B}\}$ for some invertible $S\in\mathcal{M}$. The notation $\mathcal{M}[R_{n}]$ will be used for the subalgebra of $\mathcal{M}$ consisting of those matrices whose $n$-th row is zero. Similarly, $\mathcal{M}[R_{n},C_{n}]$ indicates that the $n$-th row and $n$-th column are zero, etc. For $1\leq i,j\leq n$, let $E_{i,j}$ denote the elementary matrix in $\mathcal{M}$ with a single entry 1 in row $i$, column $j$, and [math] in each of the other $n^{2}-1$ positions. The identity matrix in $\mathcal{M}$ will be denoted by $I$. For the maximal parabolic subalgebra $\mathcal{P}:=\mathcal{M}[R_{n}]+KE_{n,n}$ in $\mathcal{M}$, define $\Omega$ to be the set of proper subalgebras $\mathcal{B}$ of $\mathcal{P}$ with $\mathcal{B}\neq\mathcal{M}[R_{n}]$.

We now describe Theorems 3.1–3.3, our main results. Theorem 3.1 shows that $\mathrm{dim\ }\mathcal{N}\leq(n-1)(n-2)$ for each $\mathcal{N}\in\Gamma$. Theorem 3.2 shows that up to similarity, $\mathcal{W}:=\mathcal{M}[R_{n},R_{n-1},C_{n}]$ and $\mathcal{W}^{\rm{T}}:=\mathcal{M}[R_{n},C_{n-1},C_{n}]$ are the only subalgebras in $\Gamma$ having maximum dimension $(n-1)(n-2)$. In Theorem 3.3, we show that $\mathrm{dim\ }\mathcal{B}\leq n^{2}-2n+3$ for each $\mathcal{B}\in\Omega$, and we classify all $\mathcal{B}\in\Omega$ of maximum dimension $n^{2}-2n+3$.

The proofs of our theorems depend on four lemmas which are proved in Section 2. Lemma 2.1 shows that $\mathcal{W}$ (and hence also $\mathcal{W}^{\rm{T}}$) is a nonunital intersection of dimension $(n-1)(n-2)$ when $n\geq 3$. Lemmas 2.2 and 2.3 show that $\mathrm{dim\ }\mathcal{L}\leq n(n-1)$ for any nonunital subalgebra $\mathcal{L}\subset\mathcal{M}$, and when equality holds, $\mathcal{L}$ must be similar to $\mathcal{M}[R_{n}]$ or $\mathcal{M}[C_{n}]$. (Thus if $\Lambda$ denotes the set of nonunital subalgebras $\mathcal{L}\subset\mathcal{M}$, Lemmas 2.2 and 2.3 classify those $\mathcal{L}\in\Lambda$ of maximum dimension.) Lemma 2.4 shows that if $\mathcal{U}\subset\mathcal{M}$ is a subalgebra with unity different from $I$, then some conjugate of $\mathcal{U}$ is contained in $\mathcal{M}[R_{n},C_{n}]$.

2 Lemmas

Recall the definition $\mathcal{W}:=\mathcal{M}[R_{n},R_{n-1},C_{n}]$.

Lemma 2.1**.**

For $n\geq 3$, $\mathcal{W}\in\Gamma$ and $\ \mathrm{dim\ }\mathcal{W}=(n-1)(n-2)$.

Proof.

For $n>1$, define $A\in\mathcal{M}$ by A=I+E_{n,n-1}\. Note that A^{-1}=I-E_{n,n-1}\. A straightforward computation shows that for $M\in\mathcal{M}[R_{n},C_{n}]$, the conjugate $AMA^{-1}$ is obtained from $M$ by replacing the (zero) bottom row of $M$ by the $(n-1)$-th row of $M$. Since the bottom two rows of $AMA^{-1}$ are identical, it follows that

[TABLE]

Since $\mathcal{W}=A^{-1}\mathcal{W}A$, this shows that $\mathcal{W}$ is the intersection of the unital subalgebras $A^{-1}\mathcal{M}[R_{n},C_{n}]A$ and $\mathcal{M}[R_{n},C_{n}]$. To see that $\mathcal{W}$ is nonunital, note that $E_{1,n-1}$ is a nonzero matrix in $\mathcal{W}$ for which $E_{1,n-1}W$ is the zero matrix for each $W\in\mathcal{W}$; thus $\mathcal{W}$ cannot have a right identity, so $\mathcal{W}\in\Gamma$. Finally, it follows from the definition of $\mathcal{W}$ that $\ \mathrm{dim\ }\mathcal{W}=(n-1)(n-2)$. ∎

*Remark: * The same proof shows that $\mathcal{W}\in\Gamma$ holds when the field $K$ is replaced by an arbitrary ring $R$ with $1\neq 0$. If moreover $R$ happens to be commutative, then the dimension of the algebra $\mathcal{W}$ over $R$ is well-defined [7, p. 483] and it equals $(n-1)(n-2)$.

Lemma 2.2**.**

For any nonunital subalgebra $\mathcal{L}\subset\mathcal{M}$, $\ \mathrm{dim\ }\mathcal{L}\leq n(n-1)$.

Proof.

It cannot happen that $\mathcal{L}+KI=\mathcal{M}$, otherwise $\mathcal{L}$ would be a two-sided proper ideal of $\mathcal{M}$, contradicting the fact that $\mathcal{M}$ is a simple ring [7, p. 280]. Since $\mathcal{L}+KI$ is a proper subalgebra of $\mathcal{M}$ containing the unity $I$, it follows from Agore [1, Cor. 2.6] that $\ \mathrm{dim\ }\mathcal{L}=-1+\ \mathrm{dim\ }(\mathcal{L}+KI)\leq n(n-1)$. ∎

Lemma 2.3**.**

Any nonunital subalgebra $\mathcal{L}\subset\mathcal{M}$ with $\ \mathrm{dim\ }\mathcal{L}=n(n-1)$ must be similar to either $\mathcal{M}[R_{n}]$ or $\mathcal{M}[C_{n}]=\mathcal{M}[R_{n}]^{\rm{T}}$.

Proof.

Consider the two parabolic subalgebras $\mathcal{P},\mathcal{P}^{\prime}\subset\mathcal{M}$ defined by

[TABLE]

Note that $\mathcal{P}^{\prime}$ is similar to the transpose $\mathcal{P}^{\rm{T}}$. Since $\mathcal{L}+KI$ is a proper subalgebra of $\mathcal{M}$ of dimension $n(n-1)+1$, it follows from Agore [1, Prop. 2.5] that $\mathcal{L}+KI$ is similar to $\mathcal{P}$ or $\mathcal{P}^{\prime}$, under the condition that $K$ is algebraically closed. However, Nolan Wallach [8] has proved that this condition can be dropped; see the Appendix. Thus, replacing $\mathcal{L}$ by a conjugate if necessary, we may assume that $\mathcal{L}+KI=\mathcal{P}$ or $\mathcal{L}+KI=\mathcal{P}^{\rm{T}}$. We will assume that $\mathcal{L}+KI=\mathcal{P}$, since the proof for $\mathcal{P}^{\rm{T}}$ is essentially the same. It suffices to show that $\mathcal{L}$ is similar to $\mathcal{M}[R_{n}]$ or $\mathcal{M}[C_{1}]$, since $\mathcal{M}[C_{1}]$ is similar to $\mathcal{M}[C_{n}]$.

Assume temporarily that each $L\in\mathcal{L}$ has all entries 0 in its upper left $(n-1)\times(n-1)$ corner. Then $n=2$, because if $n\geq 3$, then every matrix in $\mathcal{P}$ would have a zero entry in row 1, column 2, contradicting the definition of $\mathcal{P}$. Since $\mathcal{L}\subset\mathcal{M}_{2}[C_{1}]$ and both sides have dimension 2, we have $\mathcal{L}=\mathcal{M}_{2}[C_{1}]$, which proves the theorem under our temporary assumption.

When the temporary assumption is false, there exists $L\in\mathcal{L}$ with the entry 1 in row $i$, column $j$ for some fixed pair $i,j$ with $1\leq i,j\leq n-1$. Since $E_{i,i}$ and $E_{j,j}$ are in $\mathcal{P}=\mathcal{L}+KI$ and $\mathcal{L}$ is a two-sided ideal of $\mathcal{P}$, we have $E_{i,j}=E_{i,i}LE_{j,j}\in\mathcal{L}$. Consequently, $E_{a,b}=E_{a,i}E_{i,j}E_{j,b}\in\mathcal{L}$ for all pairs $a,b$ with $1\leq a\leq n-1$ and $1\leq b\leq n$. Therefore

[TABLE]

and since both $\mathcal{M}[R_{n}]$ and $\mathcal{L}$ have the same dimension $n(n-1)$, we conclude that $\mathcal{L}=\mathcal{M}[R_{n}]$. ∎

*Remark: * Any subalgebra $\mathcal{B}\subset\mathcal{M}$ properly containing $\mathcal{M}[R_{n}]$ must also contain $I$. To see this, note that $\mathcal{B}$ contains a nonzero matrix of the form

[TABLE]

If $c_{j}=0$ for all $j<n$, then $E_{n,n}\in\mathcal{B}$, so $I\in\mathcal{B}$. On the other hand, if $c_{j}\neq 0$ for some $j<n$, then $E_{n,n}=c_{j}^{-1}BE_{j,n}\in\mathcal{B}$, so again $I\in\mathcal{B}$.

Lemma 2.4**.**

Suppose that a subalgebra $\mathcal{U}\subset\mathcal{M}$ has a unity $e\neq I$. Then $S^{-1}\mathcal{U}S\subset\mathcal{M}[R_{n},C_{n}]$ for some invertible $S\in\mathcal{M}$.

Proof.

Let $r$ be the rank of the matrix $e$. Note that $e$ is idempotent, so by [5, p. 27], there exists an invertible $S\in\mathcal{M}$ for which $S^{-1}eS=D_{r}$, where $D_{r}$ is a diagonal matrix with entries 1 in rows 1 through $r$, and entries 0 elsewhere. Replacing $\mathcal{U}$ by $S^{-1}\mathcal{U}S$ if necessary, we may assume that $e=D_{r}$. Since $r\leq n-1$, we have

[TABLE]

∎

3 Theorems

Recall that $\Gamma$ is the set of all nonunital intersections in $\mathcal{M}$.

Theorem 3.1**.**

If $\mathcal{N}\in\Gamma$, then $\ \mathrm{dim\ }\mathcal{N}\leq(n-1)(n-2)$.

Proof.

Let $\mathcal{N}\in\Gamma$, so that $\mathcal{N}=\mathcal{U}\cap\mathcal{V}$ for some pair of unital subalgebras $\mathcal{U},\mathcal{V}\subset\mathcal{M}$. Since $\mathcal{N}$ is nonunital, one of $\mathcal{U},\mathcal{V}$, say $\mathcal{U}$, does not contain $I$. Thus $\mathcal{U}$ contains a unity $e\neq I$. Define $S$ as in Lemma 2.4. Replacing $\mathcal{U}$, $\mathcal{V}$, $\mathcal{N}$ by $S^{-1}\mathcal{U}S$, $S^{-1}\mathcal{V}S$, $S^{-1}\mathcal{N}S$, if necessary, we deduce from Lemma 2.4 that $\mathcal{U}$ is contained in $\mathcal{M}[R_{n},C_{n}]$. Since $\mathcal{N}$ is a nonunital subalgebra of $\mathcal{U}\subset\mathcal{M}[R_{n},C_{n}]$, it follows from Lemma 2.2 with $(n-1)$ in place of $n$ that $\ \mathrm{dim\ }\mathcal{N}\leq(n-1)(n-2)$. ∎

Theorem 3.2**.**

Let $n\geq 3$. Then up to similarity, $\mathcal{W}$ and $\mathcal{W}^{\rm{T}}$ are the only subalgebras of $\mathcal{M}$ in $\Gamma$ having dimension $(n-1)(n-2)$.

Proof.

By Lemma 2.1, every subalgebra of $\mathcal{M}$ similar to $\mathcal{W}$ or $\mathcal{W}^{\rm{T}}$ lies in $\Gamma$ and has dimension $(n-1)(n-2)$. Conversely, let $\mathcal{N}\in\Gamma$ with $\mathrm{dim\ }\ \mathcal{N}=(n-1)(n-2)$. We must show that $\mathcal{N}$ is similar to $\mathcal{W}$ or $\mathcal{W}^{\rm{T}}$.

We may assume, as in the proof of Theorem 3.1, that $\mathcal{N}$ is a nonunital subalgebra of $\mathcal{M}[R_{n},C_{n}]$. Let $\mathcal{L}$ be the subalgebra of $\mathcal{M}_{n-1}$ consisting of those matrices in the upper left $(n-1)\times(n-1)$ corners of the matrices in $\mathcal{N}$. Since $\mathrm{dim\ }\ \mathcal{L}=\mathrm{dim\ }\ \mathcal{N}=(n-1)(n-2)$, it follows from Lemma 2.3 that $\mathcal{L}$ is similar to $\mathcal{M}_{n-1}[R_{n-1}]$ or $\mathcal{M}_{n-1}[C_{n-1}]$. Thus $\mathcal{N}$ is similar to $\mathcal{W}=\mathcal{M}[R_{n},R_{n-1},C_{n}]$ or $\mathcal{W}^{\rm{T}}=\mathcal{M}[R_{n},C_{n-1},C_{n}]$. ∎

Recall that $\Omega$ denotes the set of proper subalgebras $\mathcal{B}\neq\mathcal{M}[R_{n}]$ in $\mathcal{P}$.

Theorem 3.3**.**

Let $\mathcal{B}\in\Omega$. Then $\mathrm{dim\ }\mathcal{B}\leq n^{2}-2n+3$. If $\mathcal{B}$ has maximum dimension $n^{2}-2n+3$, then $\mathcal{B}$ is similar to one of

[TABLE]

Proof.

Let $e\in\mathcal{M}$ denote the diagonal matrix of rank $n-1$ with entry [math] in row $n$ and entries $1$ in the remaining rows. Because $e$ is a left identity in $\mathcal{M}[R_{n}]$ and $\mathcal{B}e\subset\mathcal{M}[R_{n},C_{n}]$, it follows that $\mathcal{B}e$ is an algebra.

First suppose that $\mathcal{B}e=\mathcal{M}[R_{n},C_{n}]$. Then $\mathcal{P}=\mathcal{C}+\mathcal{D}$, where

[TABLE]

We proceed to show that $\mathcal{C}\cap\mathcal{D}$ is zero. Assume for the purpose of contradiction that there exists a nonzero matrix $B\in\mathcal{C}\cap\mathcal{D}$. Then $B\in\mathcal{B}$. We have $\mathcal{B}B=\mathcal{D}$, since the matrices in $\mathcal{B}$ have all possible submatrices in their upper left $(n-1)$ by $(n-1)$ corners. Thus $\mathcal{D}\subset\mathcal{B}\subset\mathcal{C}$, which implies that $\mathcal{M}[R_{n}]\subset\mathcal{B}$ and $\mathcal{P}=\mathcal{C}=\mathcal{B}+KE_{n,n}$. If $KE_{n,n}\subset\mathcal{B}$, then $\mathcal{B}=\mathcal{P}$, and if $KE_{n,n}$ is not contained in $\mathcal{B}$, then $\mathcal{B}=\mathcal{M}[R_{n}]$; either case contradicts the fact that $\mathcal{B}\in\Omega$.

Since $\mathcal{C}\cap\mathcal{D}$ is zero,

[TABLE]

Thus

[TABLE]

so the desired upper bound for $\mathrm{dim\ }\mathcal{B}$ holds when $\mathcal{B}e=\mathcal{M}[R_{n},C_{n}]$.

Next suppose that $\mathcal{B}e$ is a proper subalgebra of $\mathcal{M}[R_{n},C_{n}]$. We proceed to show that

[TABLE]

by showing that

[TABLE]

for every proper subalgebra $\mathcal{L}$ of $\mathcal{M}_{n-1}$. If $\mathcal{L}$ is nonunital, then

[TABLE]

by Lemma 2.2 (with $n-1$ in place of $n$). If $\mathcal{L}$ contains a unit different from the identity of $\mathcal{M}_{n-1}$, then by Lemma 2.4 (with $\mathcal{L}$ in place of $\mathcal{U}$),

[TABLE]

If $\mathcal{L}$ contains the identity of $\mathcal{M}_{n-1}$, then by Agore [1, Cor. 2.6],

[TABLE]

This completes the demonstration that $d\leq(n-1)(n-2)+1$.

Let $B_{1}e,B_{2}e,\dots,B_{d}e$ be a basis for $\mathcal{B}e$, with $B_{i}\in\mathcal{B}$. Since $\mathcal{B}$ is a subspace of the vector space spanned by the $d+n$ matrices

[TABLE]

it follows that

[TABLE]

Thus the desired upper bound for $\mathrm{dim\ }\mathcal{B}$ holds in all cases.

The argument above shows that when we have the equality

[TABLE]

then

[TABLE]

Moreover, from the equality $d=\mathrm{dim\ }\mathcal{B}e=(n-1)(n-2)+1$, it follows from Agore [1, Prop. 2.5] (again appealing to the Appendix to dispense with the condition of algebraic closure) that there is an invertible matrix $S$ in the set $\mathcal{M}[R_{n},C_{n}]+E_{n,n}$ such that $S^{-1}\mathcal{B}eS$ is equal to one of

[TABLE]

Since $S^{-1}\mathcal{M}E_{n,n}S=\mathcal{M}E_{n,n}$, we achieve the desired classification of $\Omega$. ∎

4 Appendix

Let $F$ be a field of characteristic [math] with algebraic closure $\overline{F}$. Given a proper subalgebra $\mathcal{C}\subset\mathcal{M}_{n}(F)$ of maximum dimension, Agore [1, Prop. 2.5, Cor. 2.6] proved that the $\bar{F}$-span of $\mathcal{C}$ is similar over $\bar{F}$ to the $\bar{F}$-span of $\mathcal{D}$ for some parabolic subalgebra $\mathcal{D}$ of maximum dimension in $\mathcal{M}_{n}(F)$. The purpose of this Appendix is to deduce that $\mathcal{C}$ is similar over $F$ to $\mathcal{D}$.

Lemma 4.1**.**

(Wallach)* Let $\mathcal{A}$ be a subspace of $\mathcal{M}_{n}(F)$ of dimension $n-1$ such that $\mathcal{A}\otimes_{F}\overline{F}$ has basis of one of the following two forms:*

a) $x_{1}\otimes\lambda_{1},x_{2}\otimes\lambda_{1},...,x_{n-1}\otimes\lambda_{1}$, with $\lambda_{1}\in(\bar{F}^{n})^{\ast},x_{j}\in\bar{F}^{n}$ and $\lambda_{1}(x_{j})=0$,

b) $x_{1}\otimes\lambda_{1},x_{1}\otimes\lambda_{2},...,x_{1}\otimes\lambda_{n-1}$, with $\lambda_{j}\in(\bar{F}^{n})^{\ast},x_{1}\in\bar{F}^{n}$ and $\lambda_{j}(x_{1})=0.$ Then in case a) $\mathcal{A}$ is $F-$conjugate (i.e. under $\mathrm{GL}(n,F)$) to the span of the matrices $E_{i,n}$ with $i=1,...,n-1$, and in case b) $\mathcal{A}$ is $F-$conjugate to the span of the matrices $E_{n,i}$ with $i=1,...,n-1$.

Proof.

In either case, if $X,Y\in\mathcal{A}$ then $XY=0$ and $X$ has rank $1$. For $X$ of rank $1$, we have $XF^{n}=Fy$ for some $y\neq 0$. Thus there exists $\mu\in\left(F^{n}\right)^{\ast}$ with $Xz=\mu(z)y=\left(y\otimes\mu\right)(z)$ for all $z$. We conclude that $\mathcal{A}$ has a basis over $F$ of the form $X_{i}=y_{i}\otimes\mu_{i}$ for $i=1,...,n-1$.

We now assume that case a) is true (the argument for the other case is essentially the same). In case a), there exists $z\in$ $\bar{F}^{n}$ such that

[TABLE]

is linearly independent over $\bar{F}$. This implies that

[TABLE]

Thus $y_{1},...,y_{n-1}$ are linearly independent. But $0=X_{i}X_{j}=\mu_{i}(y_{j})y_{i}\otimes\mu_{j}$. Thus $\mu_{i}(y_{j})=0$ for all $j=1,...,n-1$. Let $\nu$ be a non-zero element of $\left(F^{n}\right)^{\ast}$ such that $\nu(y_{i})=0$ for all $i=1,...,n-1.$ Then $\nu$ is unique up to non-zero scalar multiple. Thus $y_{i}\otimes\nu$, $i=1,...,n-1$ is an $F$–basis of $\mathcal{A}$. Clearly there exists $g\in\mathrm{GL}(n,F)$ such that if $e_{1},...,e_{n}$ is the standard basis and $\xi_{1},...,\xi_{n}$ is the dual basis then $gy_{i}=e_{i}$ and $\nu\circ g=\xi_{n}$. This completes the proof in case a). ∎

Proposition 4.2**.**

(Wallach)* Suppose that $\mathcal{L}\subset\mathcal{M}_{n}(F)$ is a subalgebra such that $\mathcal{L}\otimes_{F}\overline{F}$  is either:*

a) conjugate to the parabolic subalgebra $\mathcal{P}_{\bar{F}}$,

b) conjugate to the parabolic subalgebra $\left(\mathcal{P}_{\bar{F}}\right)^{\rm{T}}.$

In case a) $\mathcal{L}$ is $F$–conjugate to $\mathcal{P}_{F}$. In case b) $\mathcal{L}$ is $F$–conjugate to $\mathcal{P}_{F}^{\rm{T}}$.

Proof.

We just do case a) as case b) is proved in the same way. We look upon $\mathcal{L}$ as a Lie algebra over $F$. Then Levi’s theorem [4, p. 91] implies that $\mathcal{L}=S\oplus R$ with $S$ a semi-simple Lie algebra and $R$ the radical (the maximal solvable ideal). Thus $\mathcal{L}\otimes_{F}\overline{F}$ $=S\otimes_{F}\overline{F}\oplus R\otimes_{F}\overline{F}$. Therefore $R\otimes_{F}\overline{F}$ is the radical of $\mathcal{L}\otimes_{F}\overline{F}$. If we conjugate $\mathcal{L}\otimes_{F}\overline{F}$ to $\mathcal{P}_{\bar{F}}$ via $h\in\mathrm{GL}(n,\bar{F})$, then we see that

[TABLE]

has basis $E_{i,n}\ ,\ i=1,...,n-1.$ Thus hypothesis a) of Lemma 4.1 is satisfied for $\mathcal{A}=[R,R]$. There exists therefore $g\in\mathrm{GL}(n,F)$ such that $g\mathcal{A}g^{-1}$ has basis $E_{i,n}\ ,\ i=1,...,n-1$. Assume that we have replaced $\mathcal{L}$ with $g\mathcal{L}g^{-1}$. Then $\mathcal{A}$ has basis $E_{i,n}\ ,\ i=1,...,n-1$. Since $[\mathcal{L},\mathcal{A}]\subset\mathcal{A}$ and $\mathcal{P}_{F}$ is exactly the set of elements $X$ of $\mathcal{M}_{n}(F)$ such that $[X,\mathcal{A}]\subset\mathcal{A}$, we have $\mathcal{L}\subset\mathcal{P}_{F}$. Thus $\mathcal{L}=\mathcal{P}_{F}$, as both sides have the same dimension. ∎

Acknowledgment

We are very grateful to Nolan Wallach for providing the Appendix and for many helpful suggestions.