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

In this paper, the strongly ad-nilpotent elements of the Lie algebra t ( n, C ) of upper triangular complex matrices are studied. We prove that all the nilpotent matrices in t ( n, C ) are strongly ad-nilpotent if and only if n ≤ 6. Additionally, we prove that all the elements exp ( ad x ) , x strongly ad-nilpotent generate the inner automorphism group Int t ( n, C ) .


Introduction
In this paper, all the algebras are assumed to be nite dimensional. Let L be a complex Lie algebra. Since ad y is an endomorphism of L for any y ∈ L, L is the direct sum of all the generalized eigenspaces L λ (ad y): ker(ad y − λid) m , where m is the multiplicity of λ as a root of the characteristic polynomial of ad y. When L λ (ad y) ker(ad y − λid), it is the ordinary eigenspace, denoted by E λ (ad y) instead.
De nition 1. Call an element x ∈ L strongly ad-nilpotent if there exists y ∈ L and some non-zero eigenvalue λ of ad y such that x ∈ L λ (ad y).
By the de nition, every element of the generalized eigenspaces L λ (ady) associated with the non-zero eigenvalue λ of ad y is strongly ad-nilpotent. From the fact that [L λ (ad y), L μ (ad y)] ⊂ L λ+μ (ad y), we know that a strongly ad-nilpotent element must be ad-nilpotent.
We rst introduce some notations used in this paper. Let Aut L denote the group of all the automorphisms of L and Int L denote the subgroup of Aut L generated by all exp(ad x), ad x nilpotent. Denote by N(L) the set of all strongly ad-nilpotent elements of L and by E(L) the subgroup of Int L generated by all exp(ad x), x ∈ N(L). e strongly ad-nilpotent elements and the group E(L) are important tools for the proof of the conjugacy of Cartan subalgebras; see [1]. In the case that L is semisimple, for any ad-nilpotent y ∈ L, there exist h, x ∈ L such that h, x, y span a subalgebra isomorphic to sl(2, C). Especially, we have ad h(y) −2y, which shows y is strongly ad-nilpotent. So, it is an equivalence between adnilpotency and strong ad-nilpotency in a semisimple Lie algebra. Furthermore, ε(L) Int L. If L is nilpotent, it has no non-zero strongly ad-nilpotent element, and E(L) has order one.
A Lie algebra L is said to be solvable if the derived series . .. 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 t t(n, C) be the Lie algebra of upper triangular complex n × n matrices and n n(n, C) be the subalgebra of strictly upper triangular matrices. e 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 t and the inner automorphism group Int t. In Section 4, we give a characterization of strongly ad-nilpotent elements of t. A simple su cient condition is also given to determine a strongly ad-nilpotent element. In Section 5, we prove the main results of this paper. ough the Lie algebras we discussed are over C, all the results are also valid for algebraically closed field of characteristic 0.

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

Proposition 1.
Assume that L is a Lie algebra which is not nilpotent. en, N(L) has non-zero elements.
Proof. Since L is not nilpotent, by Engel's eorem, there exists x ∈ L such that ad x is not nilpotent. So, there exists non-zero eigenvalue λ of ad x such that L λ (ad x) ≠ 0 { }, which deduces the desired result. Proof. We only need to prove the necessity. Suppose that x ∈ L λ (ad y) for some y ∈ L and λ ≠ 0. en, (ad y − λid) k x � 0 for some positive integer k.
us, (ady/λ − id) k x � 0. It follows that x ∈ L 1 (ady/λ), which completes the proof. □ A Lie algebra L is called decomposable if there exist ideals L 1 and L 2 of L such that L � L 1 ⊕ L 2 . Otherwise, L is called indecomposable. About the decomposable Lie algebra, we have the following.

Proposition 2.
Let L be a Lie algebra, and x ∈ L. If L is the direct sum of its ideals L 1 and L 2 , then x � x 1 + x 2 is (strongly) ad-nilpotent if and only if x 1 and x 2 are (strongly) ad-nilpotent elements of L 1 and L 2 , respectively, where Proof. First suppose that x i ∈ N(L i ) for i � 1, 2. By Lemma 1, there exists y i ∈ L i such that (ad y i − id) k i x i � 0 for some positive integer k i . en, we get from the fact [L 1 , L 2 ] � 0. erefore, x 1 + x 2 is a strongly adnilpotent element of L. Conversely, let x � x 1 + x 2 be a strongly ad-nilpotent element of L. ere exists y ∈ L and some positive integer k such that x ∈ ker(ad y − λid) k . Write y � y 1 + y 2 , where y i ∈ L i , i � 1, 2. en, us, for i � 1, 2, we have (ad Applying by a similar argument as above, we get that x � x 1 + x 2 is adnilpotent if and only if x 1 and x 2 are ad-nilpotent elements of L 1 and L 2 , respectively.

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

The Cartan Decomposition of t(n, C)
Let It is easily known that where I is the identity matrix, is a Cartan subalgebra of t, i.e., a nilpotent subalgebra with self-normalizer in t. Furthermore, t has the Cartan decomposition as follows: where Here and thereafter, e ij denotes the matrix having one in the (i, j) position and zeros elsewhere. It is well known that h⊕∐ α∈Φ t α is a Borel subalgebra of sl(n, C). In general, a Borel subalgebra of a Lie algebra L is defined by the maximal solvable subalgebra of L.
Proof. If x � kI + x 0 , then ad x � ad x 0 . Since x 0 is nilpotent, ad x 0 is also nilpotent. On the other hand, set x � (x ij ). Assume x ii ≠ x jj for some i < j. We have (ad x) m e ij � x ii − x jj m e ij + terms linearly independent to e ij , (9) for m � 1, 2, . . .. So, ad x cannot be nilpotent.
is is a contradiction.

Proposition 3. e inner automorphism group Int t is given by
(10) Proof. By the above lemma, we know that Int t is just the subgroup of Aut t generated by exp(ad t x), x ∈ n. Note that Aut t is a linear Lie group with the Lie algebra Der t consisting of all derivations of t. Since ad t n ⊂ Der t, let G be the connected Lie subgroup of Aut t whose Lie algebra is ad t n.
Since ad t n is a nilpotent algebra, the exponential mapping is a surjection. Hence, G � exp(ad t x)|x ∈ n . From G ⊂ Int t, by the definition of Int t, we get the desired result.
□ Remark 1. It is easy to get that Int t is isomorphic to the Lie group of the unipotent upper triangular complex matrices.

A Characterization of Strongly Ad-Nilpotent Elements
We first give some equivalent conditions to depict strongly ad-nilpotent elements of t.
e following statements are equivalent: (1) x is strongly ad-nilpotent (2) ere exists y ∈ t such that x ∈ L 1 (ad y) (3) ere exists y ∈ t such that x ∈ E 1 (ad y), i.e., [y, x] � x (4) ere exists a semisimple element y of t such that [y, x] � x
Since y s can be written as a polynomial in y without constant term, we have y s ∈ t. It is obvious that ad y � ad y s + ad y n is also the Jordan decomposition of ad y. According to the fact ad y s and ad y have the same generalized eigenspaces, we have x ∈ L 1 (ad y s ). For the reason that ad y s is diagonalizable, we get L 1 (ad y s ) � E 1 (ad y s ).  Proof. For any x ∈ N(t), there exists y such that [y, x] � x. us, x is nilpotent, and hence x ∈ n.

□
Recall the basic result about the automorphism and the strongly ad-nilpotent elements.
From the above lemma, in order to determine which ones in n are strongly ad-nilpotent, we only need to consider the equivalence classes under the acting of Aut(t). For Aut(t), the readers are referred to literatures [4,5].
Denote by t * the set of invertible matrices of t. en, t * is a Lie group with the Lie algebra t. For any a ∈ t * , we can define an automorphism of t by φ a : x↦axa − 1 , for all x ∈ t.
ere is a natural group epimorphism from t * to φ a |a ∈ t * . us, we can only consider the upper triangular nilpotent matrices under upper triangular similarity (shortly written as under t * ). It has been studied by many researchers (see [6] and the references therein); related research can be seen in [7][8][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. (1) If x and y are similar under t * , then x ii � y ii for Proof. If x � zyz − 1 for an invertible upper triangular matrix z � (z ij ) with z − 1 � (z ij ), then the (i, i) element yields Here, we have used the facts that z − 1 is upper triangular and z ii � 1/z ii . If x is semisimple, then there is an invertible matrix t and a diagonal matrix d such that x � t − 1 dt. By the QR decomposition, we have t � ur, where u is a unitary matrix and r is an upper triangular matrix. en, 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 x is similar to a diagonal matrix under t * . According to the first result of the proposition, we get the second.
We call h ∈ h a maximal element with one-eigenvalue if for any h ∈ h with E 1 (ad h) ⊂ E 1 (ad h), then h � h. Denote by D the set of all the maximal elements with oneeigenvalue.
For any x ∈ n, we can write x as x � λ∈Φ x λ . Define

Journal of Mathematics
Lemma 4. With the notations as above, the following claims hold:

Proof
(1) Take a maximal linearly independent subset λ 1 , . . . , λ s of Δ h and extend it to a basis λ 1 , . . . , λ n−1 ⊂ Φ for h * . It is apparent that there exists only one element h ∈ h such that λ i (h) � 1 for en, erefore, λ ∈ Δ h . (2) Notice that both h and h are the solution of λ(l) � 1 for all λ ∈ Δ h . Since span Δ h � h * , by the uniqueness of solution, we know h � h. (3) e "if" part is from (2) and the "only if" part is from (1).

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

Proposition 5. Let x ∈ n. If the linear equations
have at least one solution h in h, then x ∈ N(t). Particularly, if Δ(x) consists of linearly independent elements, then x ∈ N(t).

Theorem 2.
With the notations as above, we have Proof. For any x ∈ N(t), by eorem 1, we have Without loss of generality, we can assume tr(y) � 0. By erefore, x ∈ tE 1 (ad h)t − 1 .

The Main Results
Theorem 3. For n ≤ 6, we have N(t(n, C)) � n(n, C). (24) Proof. e case n � 1 is trivial. By Corollary 1, it remains to prove n ⊂ N(t). Since every upper triangular matrix is similar to a direct sum of indecomposable matrices under t * , we need only to consider the indecomposable elements. e canonical forms of all the indecomposable nilpotent matrices in n(n, C)(2 ≤ n ≤ 6) are given by [6] (list in eorem 2.1 and eorem 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 Δ(x) consists of linearly independent elements for any x in the rest matrices. By Proposition 5, we know x ∈ N(t), which completes the proof. □ Corollary 2. If n ≤ 6, then for any A ∈ n(n, C), the matrix equation has at least one solution in t(n, C).

Proposition 6. Let
en, A is not strongly ad-nilpotent in t(7, C).
Although for n > 6, N(t(n, C)) is a proper subset of n(n, C) as well as the set CI + n(n, C) of all ad-nilpotent elements of t(n, C), we have the following result.
Proof. We suppose n ≥ 2, the case n � 1 being trivial. Let Z(t) denote the center of t. Since Ker ad � Z(t) � CI, we have (Ker ad) ∩ n � 0. us, ad t n is isomorphic to n. Let U be the Lie group of the unipotent upper triangular complex matrices. en, U � exp n. Since n is nilpotent and U is simply connected, Int t is isomorphic to U. For any i < j, it is clear that e ij ∈ N(t) by Proposition 5. en, we get exp ke ij � I + ke ij , for all k ∈ C, i < j. (34) It is well known that any element of U can be written as a product of some matrices of the form I + ke ij , i < j. Hence, the subgroup of U generated by all exp x, x ∈ N(t), is just U itself.
erefore, the subgroup of Int t generated by all exp(ad x), x ∈ N(t), is Int t itself; that is, E(t) � Int t. □

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

Conflicts of Interest
e author declares that there are no conflicts of interest.