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Volume 29, Issue 1

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The maximal dimension of unital subalgebras of the matrix algebra

Ana L. Agore
  • Corresponding author
  • Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium; and “Simion Stoilow” Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
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Published Online: 2016-05-12 | DOI: https://doi.org/10.1515/forum-2015-0241

Abstract

We prove that over a field of characteristic zero the maximal dimension of a proper unital subalgebra in the n×n matrix algebra is n2-n+1 and furthermore this upper bound is attained for the so-called parabolic subalgebras. We also investigate the corresponding notion of parabolic coideals for matrix coalgebras and prove that the minimal dimension of a non-zero coideal of the matrix coalgebra n(k) is n-1.

Keywords: Matrix (co)algebra; parabolic subalgebras of matrix algebras; parabolic coideals of matrix coalgebras; maximal dimension of subalgebras

MSC 2010: 16D70; 16Z05

Introduction

The problem of finding the minimal/maximal dimension of proper subobjects (possibly satisfying some extra assumptions) of a given mathematical object is a natural one and it was studied in various settings. The first result in this direction was proved by Schur [13] and asserts that [n24]+1 is an upper bound for the dimensions of commutative subalgebras of n(). Later on, Jacobson [6] showed that the same bound is valid over any field K. An elegant and simpler proof of Schur’s theorem was given by Mirzakhani in [10]. These results allowed for the introduction of the Schur invariant in Lie algebra theory as the maximal dimension of abelian subalgebras of a given Lie algebra. A vast literature emerged from the study of this invariant which plays an important role in many aspects of Lie algebra theory, see for example [3, 11] and the references therein. To give just one example, the Schur invariant has been completely determined for semisimple Lie algebras by Malcev in [8]. In this note we address the problem of finding the maximal dimension of subalgebras of the matrix algebra n(K) over a field K of characteristic zero. Our main result is Theorem 2.4 which gives an upper bound for the dimension of subalgebras in n(K) and it proves that this maximal dimension is attained for the so-called parabolic subalgebras [16], which are the associative algebra counterpart of the well-known parabolic Lie subalgebras (see [17, 18] for more details). A subalgebra 𝒜 of n(K) is called parabolic if it is similar to an algebra consisting of all matrices having non-overlapping blocks of ni×ni matrices on the diagonal, i=1,2,,s, with non-zero entries only in these blocks or above them, where (n1,n2,,ns) is a partition of n. As a consequence of Theorem 2.4 we prove in Corollary 2.6 that the maximal dimension of a proper unital subalgebra of n(K) is equal to n2-n+1. A related result was obtained in [1] where it was proved that n(K) has no subalgebras of codimension equal to 1. A somewhat similar problem was considered recently in [3, 4] in the context of Lie algebras and in [9] for Lie algebras as well as for associative algebras. In the above mentioned papers the authors are interested in finding bounds for the dimensions of abelian subalgebras or abelian ideals.

The paper ends with a brief discussion on matrix coalgebras (see Section 1 for the definition and further details). We introduce the notion of parabolic coideals for matrix coalgebras and derive some of their features based on the duality between the matrix algebra and the matrix coalgebra. As an application, we prove that the minimal dimension of a non-zero coideal of the matrix coalgebra n(K) is n-1.

1 Preliminaries

Unless otherwise stated, all vector spaces, linear maps, algebras and coalgebras are over an arbitrary field K. In some specific cases we will assume that K is algebraically closed of characteristic zero. For a vector space V we denote by V*:=HomK(V,K) the dual of V. If S is a subset of V we set S:={vV*v(S)=0}. Similarly, if Y is a subset of V* we denote Y:={xVf(x)=0forallfY}. If X is a subspace of the vector space V then the codimension of X is dimK(V/X).

By an algebra (coalgebra) we mean an associative (coassociative), unital (counital) algebra (coalgebra) over K. If A is an algebra, radA stands for the Wedderburn–Artin radical of A. Wedderburn’s main theorem ([7, Theorem 2.17]) is a fundamental result on the structure of finite-dimensional algebras, being the associative algebra counterpart of the famous Levi decomposition of a Lie algebra: it asserts that for any finite-dimensional algebra A over a field K of characteristic zero, there exists a semisimple subalgebra S of A isomorphic to A/radA such that A=SradA. Moreover, over an algebraically closed field any semisimple finite-dimensional algebra is a direct sum of matrix algebras over the same field. We refer the reader to [7, Chapter II] for a detailed account on the Wedderburn–Artin theory. For an arbitrary integer n2 we let n(K) denote the algebra of n×n matrices over the field K while 𝒰n(K) and 𝒰n(K)¯ stand for the set of all upper triangular matrices in n(K), respectively the set of all strictly upper triangular matrices in n(K). We denote by ei,jn(K) the matrix having 1 in the (i,j)-th position and zeros elsewhere. If 𝒜 is a subset of n(K), then we denote C𝒜C-1={CAC-1A𝒜}.

If A is a finite-dimensional algebra then A* has a natural coalgebra structure called the dual coalgebra on A. In the same manner, if C is a (not necessarily finite-dimensional) coalgebra then C* inherits an algebra structure called the dual algebra on C. By n(K) we denote the matrix coalgebra, i.e. n(K)=n(K) as vector spaces with the coalgebra structure given for all i, j=1,2,,n by

Δ:n(K)n(K)n(K),Δ(ei,j)=k=1nei,kek,j,ε:n(K)K,ε(ei,j)=δi,j.

It is well known that n(K)* and n(K) are isomorphic as coalgebras. Similarly, n(K)* and n(K) are isomorphic as algebras. For all unexplained notions or results from the theory of coalgebras we refer the reader to [2, 15].

We also make use of a well-known result on the dimension of subspaces of nilpotent matrices which first appeared in [5]. An improved version of this result appears in [14], where the cardinality assumption on the field K is removed. More precisely, the result in [14] states that if 𝒮 is a subspace of the vector space of n×n matrices over an arbitrary field K and 𝒮 consists of nilpotent matrices, then the maximal dimension of 𝒮 is n(n-1)2. Moreover, if equality holds then 𝒮 is conjugate to 𝒰n(K)¯.

2 Parabolic subalgebras of matrix algebras

By analogy with Lie algebra theory (see [17]), a proper subalgebra 𝒜 of the matrix algebra n(K) is called parabolic [16] if it is similar to an algebra which contains 𝒰n(K), i.e. there exists an invertible matrix U such that 𝒰n(K)U𝒜U-1. As we will see, this concept will play an important role in determining the maximal dimension of a subalgebra in a matrix algebra. It was proved in [16] that 𝒜 is a parabolic subalgebra of n(K) if and only if there exists a set of positive integers n1, n2,,ns with i=1sni=n such that 𝒜 is similar to the algebra of all matrices having non-overlapping blocks of ni×ni matrices on the diagonal with non-zero entries only in these blocks or above them. Moreover, it is straightforward to see that dimA=n22+i=1sni22.

A parabolic subalgebra 𝒜 of n(K) determined by the set of positive integers n1, n2,,ns will be called a parabolic subalgebra of type (n1,n2,,ns). If s=2 then 𝒜 will be called a maximal parabolic subalgebra.

Our next result shows that a maximal parabolic subalgebra is in fact a maximal proper subalgebra of n(K).

The maximal parabolic subalgebras are maximal proper subalgebras of Mn(K).

Proof.

Consider 𝒜 to be a parabolic subalgebra having on the diagonal two blocks U and V of dimensions l×l and respectively (n-l)×(n-l) with l1 and assume there exists a subalgebra 𝒯 of n(K) such that 𝒜𝒯, 𝒜𝒯. The elements of 𝒯 are linear combinations of the matrix units ei,j and there must be an element x𝒯-𝒜 that has a non-zero entry α in a position (i,j) with i>l and jl. Then ei,j𝒯 since ei,j=α-1ei,ixej,j. Since i>l, em,i𝒜 for all m and hence em,j=em,iei,j belongs to 𝒯 for all m. Similarly ej,k𝒜 for all k and so ei,k=ei,jej,k𝒯 for all k. Thus if ei,j belongs to 𝒯, then so do all the matrix units from row i and column j. Using the same reasoning for all these elements we can conclude that 𝒯=n(K) which finishes the proof. ∎

The parabolic subalgebras of 3(K) are similar to the following subalgebras:

(KKK0KK00K),(KKKKKK00K),(KKK0KK0KK).

The last two algebras are maximal proper subalgebras of 3(K).

If K is an algebraically closed field of characteristic zero and 𝒜 is a finite-dimensional K-algebra then by Wedderburn’s theorem [7], 𝒜 has a semisimple part which is a direct sum of matrix algebras over K of dimensions n12, n22,,ns2. In order for 𝒜 to be a subalgebra of n(K) we need to have i=1snin. However, since we are trying to maximize the dimension of 𝒜 we will assume that i=1sni=n.

Let K be an algebraically closed field of characteristic zero and A a subalgebra of Mn(K) whose semisimple part S is a direct sum of matrix algebras over K of dimensions n12, n22,,ns2 with i=1sni=n. Then dimAn22+i=1sni22. Moreover, if equality holds then A is a parabolic subalgebra of Mn(K).

Proof.

By Wedderburn’s main theorem [7, Theorem 2.17] we have 𝒜=Srad𝒜. Let dim𝒜=d and let dimrad𝒜=r. It follows that i=1sni2+r=d. Remark that each ni×ni matrix subalgebra of S contains its nilpotent subalgebra of strictly upper triangular matrices. This implies that 𝒜 contains, all together, a subalgebra of nilpotent matrices of dimension i=1sni(ni-1)2+r. Using Gerstenhaber’s result [5] on subspaces of nilpotent matrices we must have

i=1sni(ni-1)2+rn(n-1)2.

Since i=1sni2+r=d we get

dn(n-1)2+i=1sni(ni+1)2.

The conclusion now follows by using i=1sni=n.

Suppose now that equality holds. It follows that r=n22-i=1sni22 and so 𝒜 contains a subalgebra of nilpotent matrices of dimension n(n-1)2. Gerstenhaber’s result implies that any such subspace of n(K) is conjugate to 𝒰n(K)¯. Since for any invertible matrix Cn(K) the map u:𝒜C𝒜C-1 which takes any A𝒜 to CAC-1 is an algebra isomorphism we may assume without loss of generality that 𝒜 contains 𝒰n(K)¯. However, by looking at dimensions one sees that 𝒰n(K)¯ is not all of 𝒜. Now since the semisimple part of 𝒜 is a direct sum of matrix algebras of dimensions n1, n2,,ns with i=1sni=n it follows that each eii𝒜 for all i=1,2,,n. Therefore, 𝒜 contains 𝒰n(K) and the proof is now finished. ∎

Let K be an algebraically closed field of characteristic zero. Then the proper subalgebras of maximum dimension in Mn(K) are the parabolic subalgebras of type (1,n-1) and respectively (n-1,1).

Proof.

By Theorem 2.4 we know that the upper bound for the dimension of a subalgebra in n(K) is attained when the subalgebra is parabolic. In what follows we will prove that this bound is maximal precisely when the subalgebra is parabolic of type (1,n-1) and respectively (n-1,1). Since the dimension of a parabolic subalgebra of n(K) is n22+i=1sni22 in order to maximize its dimension we need to have i=1sni2 as large as possible. We denote by S:=n12+n22++ni2+(n-n1-n2--ni)2 and by S:=n12+n22++ni-12+(n-n1-n2--ni-1)2, where nt are positive integers for all t=1,2,,i, such that n1+n2++ni<n. The proof will be finished once we show that S<S. Indeed, this follows by noticing that S=S-2ni(n-n1-n2--ni)<S. Therefore we need to consider parabolic subalgebras of type (l,n-l), with l1. Then i=1sni2=l2+(n-l)2. Now it can easily be seen that for 1<l<n-1 we have l2+(n-l)2<1+(n-1)2 and the conclusion follows. ∎

Let K be a field of characteristic zero. Then the maximal dimension of a proper subalgebra of the matrix algebra Mn(K) is n2-n+1.

Proof.

If K is an algebraically closed field of characteristic zero then the assertion follows from Proposition 2.5. We will prove that the algebraically closed assumption on the field K can be dropped by simply extending the coefficients to the algebraic closure K¯ of K. Indeed, let A be a K-subalgebra of n(K). We have

K¯K¯KKK¯KAK¯Kn(K)n(K¯),

where the last isomorphism follows from [12, Lemma 7.130]. Therefore, K¯KA is a K¯-subalgebra of n(K¯) and since dimK(A)=dimK¯(K¯KA) the conclusion follows. ∎

We end the paper by looking at the matrix coalgebra case. It is well known that Xn(K) is a coideal if and only if X is a subalgebra of n(K) (see [15, Proposition 1.4.6] for a more general statement). In light of this bijection we introduce the following:

A coideal Xn(K) will be called parabolic if X is a parabolic subalgebra of n(K). If X is a parabolic subalgebra of type (n1,n2,,ns) then X will be called a parabolic coideal of type (n1,n2,,ns) as well. If s=2 then X will be called a minimal parabolic coideal.

The parabolic coideals of n(K) can be characterized as follows:

Let X be a parabolic coideal of type (n1,n2,,ns) of the matrix coalgebra Mn(K). Then X is the coideal of all matrices having non-overlapping blocks of ni×ni matrices on the diagonal with non-zero entries only below these blocks.

Proof.

If 𝒜 is a subalgebra of n(K)*n(K) (isomorphism of algebras) then 𝒜 is a coideal in n(K). The conclusion follows by a straightforward computation. ∎

The minimal parabolic coideals are minimal proper coideals of Mn(K).

Proof.

Let X be a minimal parabolic coideal of type (l,n-l) and assume there exists a coideal Y of n(K) such that YX, YX. Then XY and X is a maximal parabolic subalgebra of n(K). Using Proposition 2.2 we obtain Y=n(K). This implies Y={0} and the proof is finished. ∎

Let K be an algebraically closed field of characteristic zero. Then the non-zero coideals of minimal dimension in Mn(K) are those parabolic coideals X for which X is a parabolic subalgebra of type (1,n-1) and respectively (n-1,1).

Proof.

For any finite-dimensional vector space V, and any subspace X of V* we have dimKX=dimKV*/X. Therefore, for any coideal X in n(K) we have dimKX=dimKn(K)/X. Collaborating this result with the bijection between the coideals of n(K) and the subalgebras of n(K) (see [15]), it follows that a coideal X in n(K) has maximal codimension precisely when the subalgebra X of n(K) has maximal dimension. The conclusion now follows by Proposition 2.5. ∎

Let K be a field of characteristic zero. Then the minimal dimension of a non-zero coideal in Mn(K) is n-1.

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About the article


Received: 2015-11-26

Published Online: 2016-05-12

Published in Print: 2017-01-01


The author is Postdoctoral Fellow of Fonds Wetenschappelijk Onderzoek – Flanders (Belgium) (F.W.O. Vlaanderen). This research is part of the grant no. 88/05.10.2011 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI.


Citation Information: Forum Mathematicum, Volume 29, Issue 1, Pages 1–5, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0241.

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