Bulletin T.CXXXI de l’Académie serbe des sciences et des arts − 2005 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 30

ON DERIVATIONS OVER RINGS OF TRIANGULAR MATRICES

A. L. BARRENECHEA, C. C. PEÑA

(Presented at the 3rd Meeting, held on April 22, 2005)

A b s t r a c t. The purpose of this article is to describe the structure of derivations over general rings of upper or lower triangular matrices. Some examples are developed, together with a simple formula that gives the dimension of the module of derivations on the center of the underlying ring.

AMS Mathematics Subject Classification (2000): 16W25 - 46L57 Key Words: Derivations. Inner derivations. Lie brackets. General semi-

group ring of a group on a ring. Central simple k - algebras.

1. Introduction

The determination of the structure of derivations is relevant to develop their behaviour as operators. In general it is difficult if not possible to determine such structure. Some advances can be obtained in algebras of infinite matrices corresponding to Hilbert-Schmidt operators acting on a separable Hilbert space (cf. [2]). For non existence theorems of bounded derivations in Banach algebras of weighted sequences see [3]. For examples of general derivations in non C^{∗} algebras nor non von Neumann algebras see [4]. In some sense the lack of knowledge of such results is overtaken by appealing to alternative construccions. For instance, in the theory of C^{∗} - algebras J. Glimm (cf. [8]) introduced the notion of uniformly hyperfinite algebras (UHF - algebras), i.e., C^{∗} - unitary algebras 𝔘 endowed with an increasing sequence {𝔘_{n}} of finite dimensional full matrix algebras containing the identity whose union is dense in 𝔘. The study of ∗ - derivations defined on ∪𝔘_{n} includes the study of general quantum lattice systems. Among other successful contributions on UHF algebras and normal ∗ - derivations we mention that of S. Sakai and H. Araki (cf. [10], [1]). The corresponding notion of UHF - algebras for VN algebras goes back to F. J. Murray and J. Von Neumann (cf. [9]). J. Dixmier considered inductive limits of matrix algebras without the demand that the embeddings preserve units (cf. [7]). Latter, O. Bratteli introduced the approximately finite dimensional C^{∗} - algebras (or AF algebras), i.e., C^{∗} - algebras 𝔔 that have an increasing sequence of finite dimensional ∗ - subalgebras {𝔔_{n}} whose union is dense in 𝔔 (cf. [5]). The resource to consider the structure and properties of operators acting on finite matrix algebras is plainly relevant. Throughout this article ℜ will be a ring and _{ℜ}ℜ and ℜ_{ℜ} will denote the left and right module structures of ℜ over itself. If n is a positive integer, we shall be concerned with the ring 𝕌𝕋_{n} (ℜ) of upper triangular n × n matrices over ℜ. Of course, our conclusions hold under suitable modifications to the ring 𝕃𝕋_{n} (ℜ) of lower triangular n × n on ℜ. 𝕌𝕋_{n} (ℜ) inherits left and right module structures on ℜℜ and ℜ_{ℜ} so that

r · (x_{i,j})_{1≤i≤j≤n} = (r · x_{i,j})_{1≤i≤j≤n} or r · (x_{i,j})_{1≤i≤j≤n} = (x_{i,j} · r)_{1≤i≤j≤n}

whenever r ∈ ℜ and (x_{i,j})_{1≤i≤j≤n} ∈ 𝕌𝕋_{n} (ℜ). Let us denote these structures by 𝕌𝕋_{n} (_{ℜ}ℜ) and 𝕌𝕋_{n} (ℜ_{ℜ}) respectively. Since 𝕌𝕋_{n} (_{ℜ}ℜ) ≈ [𝕌𝕋_{n} (ℜ_{ℜ})]^{op} we´ ll restrict our research to 𝕌𝕋_{n} (_{ℜ}ℜ) . So let D (𝕌𝕋_{n} (_{ℜ}ℜ)) be the set of left derivations on 𝕌𝕋_{n} (_{ℜ}ℜ), i.e., Δ ∈ D (𝕌𝕋_{n} (_{ℜ}ℜ)) iff Δ ∈ Hom (_{ℜ}𝕌𝕋_{n} (_{ℜ}ℜ)) and Δ (η · λ) = Δ (η)·λ+η·Δ (λ) if η, λ ∈ 𝕌𝕋_{n} (ℜ) (Leibnitz rule, henceforth abbreviated by Lr). If 1 ≤ k ≤ h ≤ n we´ ll write e_{k,h} = (δ_{i,j}^{k,h})_{1≤i≤j≤n} ,

where δ_{i,j}^{k,h} are the usual Kronecker´ symbols. In general 𝕌𝕋_{n} (ℜ) is not an ℜ - algebra, but the relations r · (e_{i,j} · e_{k,h}) = (r · e_{i,j}) · e_{k,h} = e_{i,j} · (r · e_{k,h}) hold for if r ∈ ℜ and all e_{i,j}, e_{k,h} ś in 𝕌𝕋_{n} (ℜ) . In Section 2 weshall exhibit the structure of all derivations on 𝕌𝕋_{2} (_{ℜ}ℜ) , and the particular cases of ℤ_{2}, the quaternionic ring on ℝ and of a general semigroup ring of a group on a ring are considered. In Section 3, if n is a fixed positive integer in Th. 5 we develop a general structure theorem of derivations on 𝕌𝕋_{n} (_{ℜ}ℜ) . Besides, in Prop. 7 it is evaluated the dimension of D (𝕌𝕋_{n} (_{ℜ}ℜ)) as a Z(ℜ) module.

2. Derivations on 𝕌𝕋_{2} (_{ℜ}ℜ)

Proposition 1. If ℜ is an integral domain and Δ ∈ D (𝕌𝕋_{2} (_{ℜ}ℜ)) there are unique a, b ∈ Z (ℜ) so that

Δ (x · e_{1,1} + y · e_{1,2} + z · e_{2,2}) = [(x − z) · a + y · b] · e_{1,2}, x, y, z ∈ ℜ. (1)

P r o o f. Let Δ (e_{r,s}) = (ϱ_{i,j}^{r,s})_{1≤i≤j≤2} , 1 ≤ r ≤ s ≤ 2. Since e_{1,1}^{2} =

e_{1,1} by Lr we obtain ϱ_{1,1}^{1,1} = ϱ_{2,2}^{1,1} = 0. So there is a unique a ∈ ℜ so that Δ (e_{1,1}) = a · e_{1,2}. Since e_{1,2} = e_{1,1} · e_{1,2} = e_{1,2} · e_{2,2} by Lr we get ϱ_{2,2}^{1,2} = 0 and ϱ_{1,1}^{1,2} = ϱ_{2,2}^{2,2} = 0 respectively. Hence there is a unique b ∈ ℜ so that Δ (e_{1,2}) = b · e_{1,2}. Indeed, e_{2,2}^{2} = e_{2,2} and by Lr ϱ_{1,1}^{2,2} = 0, i.e., Δ (e_{2,2}) = c · e_{1,2} for a unique c ∈ ℜ. Now (1) follows because Δ ∈ Hom (_{ℜ}𝕌𝕋_{n} (_{ℜ}ℜ)) . Moreover, if η, λ ∈ 𝕌𝕋_{n} (ℜ) by (1) we can write

Δ (η · λ) = [η_{1,1} · λ_{1,1} · a + (η_{1,1} · λ_{1,2} + η_{1,2} · λ_{2,2}) · b + η_{2,2} · λ_{2,2} · c] · e_{1,2}.

(2)

On the other hand, by (1) is

Δ (η) · λ + η · Δ (λ) = [(η_{1,1} · a + η_{1,2} · b + η_{2,2} · c) · λ_{2,2}+  (3)
+η_{1,1} · (λ_{1,1} · a + λ_{1,2} · b + λ_{2,2} · c)] · e_{1,2}.

From (2) and (3) the equation

η_{1,1}·(a · λ_{2,2} + λ_{2,2} · c) = η_{1,2}·(λ_{2,2} · b − b · λ_{2,2})+η_{2,2}·(λ_{2,2} · c − c · λ_{2,2}) (4)

holds for all η_{1,1}, η_{1,2}, η_{2,2}, λ_{2,2} in ℜ. In particular, by (4) if η_{1,1} = η_{2,2} = 0, η_{1,2} = 1 and λ_{2,2} is arbitrary then b ∈ Z (ℜ) . If η_{1,1} = η_{1,2} = 0, η_{2,2} = 1 and λ_{2,2} is arbitrary then c ∈ Z (ℜ) . If η_{1,2} = η_{2,2} = 0 and η_{1,1} = λ_{2,2} = 1 then a + c = 0, i.e a ∈ Z (ℜ). Now, (1) follows by (2) since c = −a. □

Example 2. D (𝕌𝕋_{2} (ℤ_{2})) = {0, Δ_{1}, Δ_{2}, Δ_{3}} , where

\begin{aligned}Δ_{1} (x · e_{1,1} + y · e_{1,2} + z · e_{2,2}) & = y · e_{1,2}, \\ Δ_{2} (x · e_{1,1} + y · e_{1,2} + z · e_{2,2}) & = (x + z) · e_{1,2}, \\ Δ_{3} (x · e_{1,1} + y · e_{1,2} + z · e_{2,2}) & = (x + y + z) · e_{1,2}. \end{aligned}

Example 3. If ℍ (ℝ) denotes the ring of real quaternions D (𝕌𝕋_{2} (_{ℍ(ℝ)}ℍ (ℝ))) consists of those elements Δ = Δ(a, b) with a, b ∈ ℝ and

Δ (a, b) (x · e_{1,1} + y · e_{1,2} + z · e_{2,2}) = [(x − z) · a + y · b]·e_{1,2}, x, y, z ∈ ℍ (ℝ) .

Example 4. If 𝔊 is a group and ℜ is a ring let ℜ [𝔊] be the semigroup ring of 𝔊 over ℜ. Elements of ℜ [𝔊] are functions 𝔊 → ℜ of finite support and

(a + b)(x) = a(x) + b(x), (a · b) (x) = ∑_{u,v∈𝔊:u·v=x} a (u) · b (v)

if a, b ∈ ℜ [𝔊], x ∈ 𝔊. ℜ [𝔊] becomes a non abelian ring and D (𝕌𝕋_{2} (_{ℜ[𝔊]}ℜ [𝔊])) contains elements of the form (1) so that a, b ∈ Z(ℜ) [𝔊] that preserve conjugacy classes, i.e., a (x) = a (u · x · u^{−1}), b (x) = b (u · x · u^{−1}) if u, x ∈ 𝔊.

3. A structure theorem and dimensionality

Theorem 5. A left homomorphism Δ on 𝕌𝕋_{n} (ℜ) is a derivation if and only if there is a unique set of scalars \{ϱ_{u,v}^{k,h}\}_{{1≤k≤h≤n \atop 1≤u≤v≤n}} so that

(ϱ_{k,h}^{k,h})_{1≤k,h≤n} , (ϱ_{j,k}^{k,k})_{1≤j,k≤n} , (ϱ_{h,l}^{h,h})_{1≤h,l≤n} ∈ 𝕌𝕋_{n} (Z(ℜ)),  (5)

if 1 ≤ k ≤ h ≤ n then

ϱ_{u,v}^{k,h} = 0 if k < u or v < h,  (6)

ϱ_{u,v}^{k,h} = \{ \begin{matrix}ϱ_{u,k}^{k,k} & if & 1 ≤ u < k ≤ n, 1 ≤ v = h ≤ n, \\ 0 & if & \{ \begin{matrix}1 ≤ u < k ≤ h ≤ n, 1 ≤ u ≤ v ≤ n, v ≠ h, \\ or \\ 1 ≤ k ≤ h < v ≤ n, 1 ≤ u ≤ v ≤ n, u ≠ k, \end{matrix} \\ ϱ_{h,v}^{h,h} & if & 1 ≤ h < v ≤ n, 1 ≤ u = k ≤ n, \end{matrix}  (7)

\begin{aligned}ϱ_{u,k}^{u,u} + ϱ_{u,h}^{k,h} & = 0 if 1 ≤ u < k ≤ n, 1 ≤ k ≤ h ≤ n, \\ ϱ_{k,v}^{k,h} + ϱ_{h,v}^{v,v} & = 0 if 1 ≤ h < v ≤ n, 1 ≤ k ≤ h ≤ n. \end{aligned}  (8)

and

ϱ_{k,h}^{k,h} = ϱ_{k,l}^{k,l} + ϱ_{l,h}^{l,h} if 1 ≤ k ≤ l ≤ h ≤ n.  (9)

Thus, if μ ∈ 𝕌𝕋_{n} (ℜ) then

Δ (μ) = ∑_{1≤k<h≤n} (μ_{k,h} · ϱ_{k,h}^{k,h} − ∑_{k<l≤h} μ_{l,h} · ϱ_{k,l}^{k,k} − ∑_{k≤l<h} μ_{k,l} · ϱ_{l,h}^{h,h}) · e_{k,h}.

P r o o f. For 1 ≤ i ≤ j ≤ n is Δ (e_{i,j}) = ∑_{1≤u≤v≤n} ϱ_{u,v}^{i,j} · e_{u,v} for some unique ϱ_{u,v}^{i,j} ś. Since e_{i,i}^{2} = e_{i,i} by Lr is ϱ_{u,v}^{i,j} = 0 if u ≠ i and v ≠ i. Indeed, it is easily seeing that ϱ_{i,i}^{i,i} are necessarily zero in this case and so

Δ (e_{i,i}) = ∑_{u=1}^{i−1} ϱ_{u,i}^{i,i} · e_{u,i} + ∑_{v=i+1}^{n} ϱ_{i,v}^{i,i} · e_{i,v}, 1 ≤ i ≤ n.  (11)

In particular, sums in (11) are assumed to be zero in case that i = 1 or i = n. Let 1 ≤ k < i ≤ n, 1 ≤ k ≤ h ≤ n. By (11) is Δ (e_{i,i}) · e_{k,h} = 0 and as e_{i,i} · e_{k,h} = 0 by Lr we get e_{i,i} · Δ (e_{k,h}) = 0. Analogously, if 1 ≤ i < h ≤ n, 1 ≤ k ≤ h ≤ n by (11) is e_{k,h} · Δ (e_{i,i}) = 0. Since e_{k,h} · e_{i,i} = 0 by Lr we get Δ (e_{k,h}) · e_{i,i} = 0. Thus we can conclude (6). If 1 ≤ k ≤ h ≤ n by (11) we write

Δ (e_{k,h}) = ∑_{u=1}^{k−1} ϱ_{u,k}^{k,k}·e_{u,h}+∑_{v=k}^{n} ϱ_{k,v}^{k,h}·e_{k,v} = ∑_{u=1}^{h} ϱ_{u,h}^{k,h}·e_{u,h}+ ∑_{v=h+1}^{n} ϱ_{h,v}^{h,h}·e_{k,v}. (12)

By (6) and (12) it follows (7). If 1 ≤ u < k ≤ n, 1 ≤ k ≤ h ≤ n by (11) is

Δ (e_{u,u}) · e_{k,h} = ϱ_{u,k}^{u,u} · e_{u,h} = −e_{u,u} · Δ (e_{k,h}) .  (13)

If 1 ≤ h < v ≤ n, 1 ≤ k ≤ h ≤ n by (11) is

e_{k,h} · Δ (e_{v,v}) = ϱ_{h,v}^{v,v} · e_{k,v} = −Δ (e_{k,h}) · e_{v,v}.  (14)

By (13) and (14) we deduce (8). Since ϱ_{k,k}^{k,k} = 0, besides (8) and (11) it follows (10) for μ = e_{k,k} if 1 ≤ k ≤ n. If 1 ≤ k < h ≤ n then (10) holds for μ = e_{k,h} by (6), (7) and (8). Now, the general case in (10) holds since ϱ_{k,k}^{k,k} = 0 if ≤ k ≤ n, (6), (7) and (8). Moreover, if η, λ ∈ 𝕌𝕋_{n} (ℜ) and 1 ≤ k ≤ h ≤ n by Lr the following identities hold

\begin{matrix}∑_{c=k}^{h} η_{k,c} · λ_{c,h} · ϱ_{k,h}^{k,h} − ∑_{a=k+1}^{h} ∑_{b=a}^{h} η_{a,b} · λ_{b,h} · ϱ_{k,a}^{k,k} − ∑_{b=k}^{h−1} ∑_{a=k}^{b} η_{k,a} · λ_{a,b} · ϱ_{b,h}^{h,h} \\ = ∑_{b=k}^{h} [(η_{k,b} · ϱ_{k,b}^{k,b} − ∑_{a=k+1}^{b} η_{a,b} · ϱ_{k,a}^{k,k} − ∑_{a=k}^{b−1} η_{k,a} · ϱ_{a,b}^{b,b}) · λ_{b,h}+ \\ + η_{k,b} · (λ_{b,h} · ϱ_{b,h}^{b,h} − ∑_{a=b+1}^{h} λ_{a,h} · ϱ_{b,a}^{b,b} − ∑_{a=b}^{h−1} λ_{b,a} · ϱ_{a,h}^{h,h})]. \end{matrix} (15)

82 A. L. Barrenechea, C. C. Peña

If 1 ≤ k ≤ h ≤ n and η = e_{k,h} in (15) then λ_{h,h} · ϱ_{k,h}^{k,h} = ϱ_{k,h}^{k,h} · λ_{h,h}. Since λ is arbitrary then (ϱ_{k,h}^{k,h})_{1≤k≤h≤n} ∈ 𝕌𝕋_{n} (Z(ℜ)). If 1 ≤ k < a, r ∈ ℜ, on choosing h ≥ a we write η = e_{a,a} and λ = r·e_{a,h}. So, by (15) is r·ϱ_{k,a}^{k,k} = ϱ_{k,a}^{k,k}·r and since r is arbitrary and ϱ_{a,a}^{a,a} = 0 is (ϱ_{k,a}^{k,k})_{1≤k≤a≤n} ∈ 𝕌𝕋_{n} (Z(ℜ)). Finally, by (8) is (ϱ_{k,a}^{a,a})_{1≤k≤a≤n} ∈ 𝕌𝕋_{n} (Z(ℜ)) and we are ready to prove (9). Indeed, we have

∑_{a=k+1}^{h} ∑_{b=a}^{h} η_{a,b} · λ_{b,h} · ϱ_{k,a}^{k,k} = ∑_{b=k+1}^{h} ( ∑_{a=k+1}^{b} η_{a,b} · ϱ_{k,a}^{k,k}) · λ_{b,h},  (16)

\begin{aligned}∑_{b=k}^{h−1} ∑_{a=k}^{b} η_{k,a} · λ_{a,b} · ϱ_{b,h}^{h,h} & = ∑_{b=k}^{h−1} η_{k,b} ∑_{a=b}^{h−1} λ_{b,a} · ϱ_{a,h}^{h,h}, \\ ∑_{b=k}^{h} & (∑_{a=k}^{b−1} η_{k,a} · ϱ_{a,b}^{b,b} · λ_{b,h}+ \\ + ∑_{a=b+1}^{h} η_{k,b} · λ_{a,h} · ϱ_{b,a}^{b,b}) & = ∑_{a=k}^{h} ∑_{a=b+1}^{h} η_{k,a} · λ_{b,h} · (ϱ_{a,b}^{a,a} + ϱ_{a,b}^{b,b}) = 0. \end{aligned}

Thus by (15) and (16)

(∑_{c=k}^{h} η_{k,c} · λ_{c,h}) · ϱ_{k,h}^{k,h} = ∑_{c=k}^{h} η_{k,c} · λ_{c,h} · (ϱ_{k,c}^{k,c} + ϱ_{c,h}^{c,h})

and (9) follows if we put η = e_{k,c} and λ = e_{c,h} for each c ∈ {k, k + 1, ..., h} . By the same reasoning, if \{ϱ_{u,v}^{k,h}\}_{{1≤k≤h≤n \atop 1≤u≤v≤n}} is a subset of ℜ so that (5), (6), (7), (8) and (9) hold then (10) defines an element μ ∈ D (𝕌𝕋_{n} (ℜ)) . □

Remark 6. By (8) in Th. 5 is ϱ_{k,h}^{k,k} + ϱ_{k,h}^{h,h} = 0 if 1 ≤ k ≤ h ≤ n. By (9) the matrix (ϱ_{k,h}^{k,h})_{1≤k≤h≤n} is determinated by ϱ_{i,i+1}^{i,i+1}, 1 ≤ i < n. Indeed, we get the following

Proposition 7. dim_{Z(ℜ)} D (_{ℜ}𝕌𝕋_{n} (ℜ)) = (n^{2} + n − 2) /2.

P r o o f. If 1 ≤ j < n, μ ∈ 𝕌𝕋_{n} (ℜ) let Δ_{j} (μ) = ∑_{1≤k≤n−j<h≤n} μ_{k,h}·e_{k,h}. Clearly each Δ_{j} ∈ Hom (_{ℜ}𝕌𝕋_{n} (_{ℜ}ℜ)) . Indeed, if 1 ≤ k ≤ n − j < h ≤ n and η, λ ∈ 𝕌𝕋_{n} (ℜ) then

\begin{aligned}(Δ_{j} (η) · λ + η · Δ_{j} (λ))_{k,h} & = ∑_{l=n−j+1}^{h} η_{k,l} · λ_{l,h} + ∑_{l=k}^{n−j} η_{k,l} · λ_{l,h} \\ & = (η · λ)_{k,h} = Δ_{j} (η · λ)_{k,h} . \end{aligned}

Since Δ_{j} (η · λ)_{k,h} = (Δ_{j} (η) · λ)_{k,h} = (η · Δ_{j} (λ))_{k,h} = 0 if n − j < k or h ≤ n − j each Δ_{j} becomes a derivation. Now, if 1 ≤ k < h ≤ n and μ ∈ 𝕌𝕋_{n} (ℜ) we´ ll write Λ_{k,h} (μ) = [μ, e_{k,h}] = μ · e_{k,h} − e_{k,h} · μ, i.e., Λ_{k,h} is the inner derivation defined by e_{k,h} and the Lie bracket [·, ·] . We´ ll prove that the set B = {Δ_{j}}_{1≤j<n} ∪ {Λ_{k,h}}_{1≤k<h≤n} is a base of D (_{ℜ}𝕌𝕋_{n} (ℜ)) over Z (ℜ) . For, let

∑_{j=1}^{n−1} a_{j} · Δ_{j} + ∑_{1≤k<h≤n} b_{k,h} · Λ_{k,h} = 0  (17)

for some constants a_{j} ´ s and b_{k,h} ´ s in Z (ℜ) . By (17) we obtain that

μ_{k,h} · ∑_{j=n−h+1}^{n−k} a_{j} + ∑_{l=k}^{h−1} b_{l,h} · μ_{k,l} − ∑_{m=k+1}^{h} b_{k,m} · μ_{m,h} = 0  (18)

for all μ ∈ 𝕌𝕋_{n} (ℜ) and 1 ≤ k < h ≤ n. If we put μ = e_{1,2} by (18) is a_{n−1} = 0. If μ = e_{1,l} with 2 < l ≤ n by (18) is a_{n−1} + ... + a_{n−l+1} = 0. If we already proved that a_{n−j} = 0 if 1 ≤ j < l − 1 then a_{n−l+1} = 0, i.e., we get a_{1} = ... = a_{n−1} = 0. Now, if 1 ≤ k < l < h ≤ n and μ = e_{k,l} by (18) is b_{l,h} = 0, i.e., b_{l,h} = 0 if 2 ≤ l < h ≤ n. If 1 < m ≤ h ≤ n and μ = e_{m,h} by (18) is b_{1,m} = 0, and so B is linearly independent. Finally, given Δ ∈ D (_{ℜ}𝕌𝕋_{n} (ℜ)) and 1 ≤ i ≤ j ≤ n we write as before Δ (e_{i,j}) = ∑_{1≤u≤v≤n} ϱ_{u,v}^{i,j} ·e_{u,v} for some unique ϱ_{u,v}^{i,j} ś. By Th. 5 we already know that all ϱ_{u,v}^{i,j} ś belong to Z (ℜ) . Further, by (6), (7), (8) and (9) it is readily seeing that

Δ = ∑_{j=1}^{n−1} ϱ_{n−j,n−j+1}^{n−j,n−j+1} · Δ_{j} + ∑_{1≤k<h≤n} ϱ_{k,h}^{k,k} · Λ_{k,h}  (19)

and our claim follows. □

Remark 8. As a consequence of the Skolem-Noether theorem (cf. [6], pags. 93 and 105) every k linear derivation on a finite dimensional central simple k - algebra 𝔔 is inner. This result is no longer applicable because 𝕌𝕋_{n} (_{ℜ}ℜ) is not central as a Z (ℜ) - algebra. For instance, the set of matrices μ ∈ 𝕌𝕋_{n} (_{ℜ}ℜ) whose nth row is null is a non zero bilateral ideal. However, observe that Δ_{j} (∘) = [∑_{l=1}^{n−j} e_{l,l}, ∘] if 1 ≤ j < n. Since each Λ_{k,h} is inner by (19) it follows that every derivation on 𝕌𝕋_{n} (_{ℜ}ℜ) is inner.

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UNCentro-FCExactas-NuCoMPA-Dpto. de Matemáticas Tandil, Pcia. de Bs. As. Argentina ccpenia@exa.unicen.edu.ar / analucia@exa.unicen.edu.ar