Strongly Ad-Nilpotent Elements of the Lie Algebra of Upper Triangular Matrices

Abstract

In this paper, the strongly ad-nilpotent elements of the Lie algebra of upper triangular complex matrices are studied. We prove that all the nilpotent matrices in are strongly ad-nilpotent if and only if . Additionally, we prove that all the elements , strongly ad-nilpotent generate the inner automorphism group .

1. Introduction

In this paper, all the algebras are assumed to be finite dimensional. Let be a complex Lie algebra. Since is an endomorphism of for any , is the direct sum of all the generalized eigenspaces , where is the multiplicity of as a root of the characteristic polynomial of . When , it is the ordinary eigenspace, denoted by instead.

Definition 1. Call an element strongly ad-nilpotent if there exists and some non-zero eigenvalue of such that .

By the definition, every element of the generalized eigenspaces associated with the non-zero eigenvalue of is strongly ad-nilpotent. From the fact that , we know that a strongly ad-nilpotent element must be ad-nilpotent.

We first introduce some notations used in this paper. Let denote the group of all the automorphisms of and denote the subgroup of generated by all , nilpotent. Denote by the set of all strongly ad-nilpotent elements of and by the subgroup of generated by all , .

The strongly ad-nilpotent elements and the group are important tools for the proof of the conjugacy of Cartan subalgebras; see [1]. In the case that is semisimple, for any ad-nilpotent , there exist such that span a subalgebra isomorphic to . Especially, we have , which shows is strongly ad-nilpotent. So, it is an equivalence between ad-nilpotency and strong ad-nilpotency in a semisimple Lie algebra. Furthermore, . If is nilpotent, it has no non-zero strongly ad-nilpotent element, and has order one.

A Lie algebra is said to be solvable if the derived seriessatisfies for some , where . In this paper, we consider the strongly ad-nilpotent elements of a special class of solvable Lie algebras, the linear Lie algebras of upper triangular complex matrices. Let be the Lie algebra of upper triangular complex matrices and be the subalgebra of strictly upper triangular matrices.

The paper is organized as follows. Section 2 is devoted to introducing some results about the strongly ad-nilpotent elements in general Lie algebras. In Section 3, we present the Cartan decomposition of and the inner automorphism group . In Section 4, we give a characterization of strongly ad-nilpotent elements of . A simple sufficient condition is also given to determine a strongly ad-nilpotent element. In Section 5, we prove the main results of this paper. Though the Lie algebras we discussed are over , all the results are also valid for algebraically closed field of characteristic 0.

2. Preliminaries

In this section, we introduce some results about the strongly ad-nilpotent elements for general Lie algebras.

Proposition 1. Assume that is a Lie algebra which is not nilpotent. Then, has non-zero elements.

Proof. Since is not nilpotent, by Engel’s Theorem, there exists such that is not nilpotent. So, there exists non-zero eigenvalue of such that , which deduces the desired result.

Lemma 1. Let be a Lie algebra, and . Then, if and only if there exists such that .

Proof. We only need to prove the necessity. Suppose that for some and . Then, for some positive integer . Thus, . It follows that , which completes the proof.

A Lie algebra is called decomposable if there exist ideals and of such that . Otherwise, is called indecomposable. About the decomposable Lie algebra, we have the following.

Proposition 2. Let be a Lie algebra, and . If is the direct sum of its ideals and , then is (strongly) ad-nilpotent if and only if and are (strongly) ad-nilpotent elements of and , respectively, where .

Proof. First suppose that for . By Lemma 1, there exists such that for some positive integer . Then, we getfrom the fact . Therefore, is a strongly ad-nilpotent element of .
Conversely, let be a strongly ad-nilpotent element of . There exists and some positive integer such that . Write , where . Then,Thus, for , we have , which shows .
Applyingby a similar argument as above, we get that is ad-nilpotent if and only if and are ad-nilpotent elements of and , respectively.

According to a theorem of Lan [2], a Lie algebra is solvable if and only if the set of all the ad-nilpotent elements of is a linear subspace of . Furthermore, if is solvable, then the set {x ∈ L|adx is nilpotent} is the nilpotent radical of , the maximal nilpotent ideal of ; see [3].

3. The Cartan Decomposition of

Let

It is easily known thatwhere is the identity matrix, is a Cartan subalgebra of , i.e., a nilpotent subalgebra with self-normalizer in .

Furthermore, has the Cartan decomposition as follows:where . Here, is the dual space of . Set , where for all . Then,

Here and thereafter, denotes the matrix having one in the position and zeros elsewhere. It is well known that is a Borel subalgebra of . In general, a Borel subalgebra of a Lie algebra is defined by the maximal solvable subalgebra of .

Lemma 2. Let . Then, is ad-nilpotent if and only if , where and .

Proof. If , then . Since is nilpotent, is also nilpotent. On the other hand, set . Assume for some . We havefor . So, cannot be nilpotent. This is a contradiction.

Proposition 3. The inner automorphism group is given by

Proof. By the above lemma, we know that is just the subgroup of generated by , . Note that is a linear Lie group with the Lie algebra consisting of all derivations of . Since , let be the connected Lie subgroup of whose Lie algebra is . Since is a nilpotent algebra, the exponential mapping is a surjection. Hence, . From , by the definition of , we get the desired result.

Remark 1. It is easy to get that is isomorphic to the Lie group of the unipotent upper triangular complex matrices.

4. A Characterization of Strongly Ad-Nilpotent Elements

We first give some equivalent conditions to depict strongly ad-nilpotent elements of .

Theorem 1. Let . The following statements are equivalent:(1) is strongly ad-nilpotent(2)There exists such that (3)There exists such that , i.e., (4)There exists a semisimple element of such that

Proof.   is followed from Lemma 1.  Let be the Jordan decomposition, i.e., is semisimple, is nilpotent, and . Since can be written as a polynomial in without constant term, we have . It is obvious that is also the Jordan decomposition of . According to the fact and have the same generalized eigenspaces, we have . For the reason that is diagonalizable, we get .  and are obvious.

Corollary 1. is a subset of .

Proof. For any , there exists such that . Thus, is nilpotent, and hence .

Recall the basic result about the automorphism and the strongly ad-nilpotent elements.

Lemma 3 (see [1]). Let be a Lie algebra. If , then for all and for all .

From the above lemma, in order to determine which ones in are strongly ad-nilpotent, we only need to consider the equivalence classes under the acting of . For , the readers are referred to literatures [4, 5].

Denote by the set of invertible matrices of . Then, is a Lie group with the Lie algebra . For any , we can define an automorphism of by , for all . There is a natural group epimorphism from to . Thus, we can only consider the upper triangular nilpotent matrices under upper triangular similarity (shortly written as under ). It has been studied by many researchers (see [6] and the references therein); related research can be seen in [7–9]. Note that there are only finitely upper triangular similarity classes in the size 5 or less and infinitely classes in the size 6 or higher.

Proposition 4. Let and be two elements of .(1)If and are similar under , then for (2)If is semisimple, then is similar to under

Proof. If for an invertible upper triangular matrix with , then the element yieldsHere, we have used the facts that is upper triangular and .
If is semisimple, then there is an invertible matrix and a diagonal matrix such that . By the decomposition, we have , where is a unitary matrix and is an upper triangular matrix. Then,From the above equation, the left is upper triangular and the right is normal. By the fact that an upper triangular normal matrix must be diagonal, we get that is similar to a diagonal matrix under . According to the first result of the proposition, we get the second.

For any , define

We call a maximal element with one-eigenvalue if for any with , then . Denote by the set of all the maximal elements with one-eigenvalue.

For any , we can write as . Define

Lemma 4. With the notations as above, the following claims hold:(1)Let . If , then there exists such that and .(2)Let . If and , then .(3)Let . Then, if and only if .

Proof. (1)Take a maximal linearly independent subset of and extend it to a basis for . It is apparent that there exists only one element such that for all . Obviously, . For any , there exist such that . Thus, Then, Therefore, .(2)Notice that both and are the solution of for all . Since , by the uniqueness of solution, we know .(3)The “if” part is from (2) and the “only if” part is from (1).

From the above lemma, we know is a finite set. It seems that it is an interesting question to determine the number of elements in .

Proposition 5. Let . If the linear equationshave at least one solution in , then . Particularly, if consists of linearly independent elements, then .

Proof. Fromwe see is strongly ad-nilpotent.

Example 1. Let . ConsiderThen,It is easy to obtain the unique solutionsatisfying for all . So, is a strongly ad-nilpotent element of .

Theorem 2. With the notations as above, we have

Proof. For any , by Theorem 1, we haveWithout loss of generality, we can assume . By Proposition 4, there exists such that for some . So, . By Lemma 4, there is such that . Therefore, .

5. The Main Results

Theorem 3. For , we have

Proof. The case is trivial. By Corollary 1, it remains to prove . Since every upper triangular matrix is similar to a direct sum of indecomposable matrices under , we need only to consider the indecomposable elements. The canonical forms of all the indecomposable nilpotent matrices in are given by [6] (list in Theorem 2.1 and Theorem 2.2 in [6]). In the lists, the last matrix has been proved to be strongly ad-nilpotent in Example 1. Apart from the last matrix, it is easy to check that consists of linearly independent elements for any in the rest matrices. By Proposition 5, we know , which completes the proof.

Corollary 2. If , then for any , the matrix equationhas at least one solution in .

Proposition 6. Let

Then, is not strongly ad-nilpotent in .

Proof. Assume that is strongly ad-nilpotent. By Theorem 1, there exists such thatWe can set ; otherwise, replace by for some suitable . First, comparing the elements of (27), respectively, we haveThus, . Next, from the elements of (27), we obtainwhich deduces . By (28), we get . Then, from the elements of (27), we obtainwhich forces . Therefore, by (28), we see , contrary to .

From Theorem 3 and Proposition 6, we immediately get the following main result.

Theorem 4. Let be the Lie algebra of upper triangular complex matrices, be the subalgebra of strictly upper triangular matrices, and be the set of all strongly ad-nilpotent elements of . Then,with equality holds if and only if .

Although for , is a proper subset of as well as the set of all ad-nilpotent elements of , we have the following result.

Theorem 5. For any , we have

Proof. We suppose , the case being trivial. Let denote the center of . Sincewe have . Thus, is isomorphic to . Let be the Lie group of the unipotent upper triangular complex matrices. Then, . Since is nilpotent and is simply connected, is isomorphic to .
For any , it is clear that by Proposition 5. Then, we getIt is well known that any element of can be written as a product of some matrices of the form . Hence, the subgroup of generated by all , is just itself. Therefore, the subgroup of generated by all , is itself; that is, .

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author was supported by NSFC (no. 12131012).