Characterizations of Lie -Centralizers on Certain Trivial Extension Algebras

Abstract

In this paper, we describe the structure of Lie -centralizers of a trivial extension algebra. We then present some conditions under which a Lie -centralizer on a trivial extension algebra is proper. As an application, we consider Lie -centralizers on a triangular algebra.

1. Introduction

Let be a unital commutative ring, be a unital algebra over , and be the center of . Let denote the Lie product of elements . An -linear map is called a left (right) centralizer if holds for all . Furthermore, is called a centralizer if it is both a left centralizer and a right centralizer. Centralizers on rings as well as algebras have been extensively investigated (see [1–5]). An -linear map is called a Lie centralizer if for all . It is easy to check that is a Lie centralizer on if and only if for all . If a Lie centralizer can be expressed as , where and is a linear map vanishing at commutators for all . Then, the -linear map is called a proper Lie centralizer. Recently, the structure of Lie centralizers is studied by many mathematicians (see [6–10]).

Let be a unital algebra over and be a -bimodule. Then, the direct product equipped with the pairwise addition, scalar product, and the algebra product is given bywhich forms a unital algebra, which is called a trivial extension algebra of by and will be denoted by . The center of is given by

Trivial extension algebras have been extensively studied in algebra and analysis (see [11–16]). In this paper, we will study the structure of Lie -centralizers on a trivial extension algebra.

2. Preliminaries

In this paper, we mainly discuss the trivial extension algebra for which has a nontrivial idempotent satisfyingwhere . A triangular algebra is an important example of a trivial extension algebra that satisfies (3). Let and be unital algebras over a unital commutative ring , and let be a unital -bimodule. Then, the setforms an algebra under the usual matrix addition and formal matrix multiplication. Such an algebra is called a triangular algebra (see [17]). It is easy to prove that is isomorphic to the trivial extension algebra , where the algebra is equipped with the usual pairwise operations and is regarded as an -bimodule equipped with the module operations and for all and . Assuming and , it is easy to show that is a nontrivial idempotent of and for all . In addition, we can get

If a trivial extension algebra satisfies (3), then the center of coincides with

Next, we give the definition of Lie -centralizers. Let us define the following sequence of polynomials:

The polynomial is said to be an -th commutator . A Lie -centralizer is an -linear map which satisfies the rulefor all . If there exists an element and an -linear map vanishing on each -th commutator such that for all , then the Lie -centralizer is called a proper Lie -centralizer.

Now, we state some lemmas which are very important for proving the main results.

Lemma 1. (see [13] Proposition 2.5.) Let be a unital algebra and be a -bimodule. Suppose that has a nontrivial idempotent and denote . Then, the following statements are equivalent:(i)For all , .(ii)For all , .(iii)For all , .(iv)For all and , and .

Lemma 2. Let be a unital algebra containing a nontrivial idempotent and be a -bimodule satisfying for all , where . Then, every Lie -centralizer satisfies for all .

Proof. Since is a Lie -centralizer, it follows thatfor all . According to Lemma 1, we obtain for all and . In a similar way, we get for all and . Therefore, satisfies for all .

In particular, based on the fact that every centralizer is a Lie -centralizer, if is a centralizer, then we have for all .

Lemma 3. (see [16] Lemma 2.2). Let be an -linear map and have the following form , then is a centralizer if and only if the following conditions hold:(1) is a centralizer;(2) is a centralizer;(3) for all and ;(4) and for all and ;(5) for all .

Lemma 4. (see [12], Lemma 3.11). Assume that is a trivial extension algebra satisfying (3). Then, the following statements hold:(1)The center of is given by(2), if one of the following conditions holds:(i) and is faithful as a right -module.(ii) and is faithful as a left -module.

Lemma 5. An -linear map is a centralizer if and only if there exists an element such that for all .

Proof. Suppose that is a centralizer, then we havefor all . Set , then we get and for all .
Conversely, it is clear.

3. Lie -Centralizers on

The following result gives the structure of a Lie -centralizer on a trivial extension algebra.

Theorem 6. Let be an -linear map and have the following form:where are -linear maps. Then, is a Lie -centralizer if and only if the following conditions are satisfied:(i) is a Lie -centralizer;(ii) is a Lie -centralizer;(iii) and for all , , and ;(iv) and for all , , and ;(v) for all , , and .

Proof. Since is a Lie -centralizer on , it follows thatfor all .
Let us choose in (13). Then, we obtainfor all . Comparing the above equations, we have that and are Lie -centralizers.
Let us consider in (13). Then, we deducefor all . Comparing the abovementioned relations, we concludeSimilarly, considering in (13), , we findfor all , where . Setting in (13), we getSince , it follows from (17) and (18) thatfor all .
If we take in (13), then we arrive atHence, for all . In an analogous way, we obtain for all , where .
Conversely, taking , we get from (i)–(v) thatHence, is a Lie -centralizer on .

Now, we can present the first main result of this paper, which provides the necessary and sufficient conditions for a Lie -centralizer on a trivial extension algebra satisfying (3) to be proper.

Theorem 7. Let be a trivial extension algebra satisfying (3). Suppose that is a Lie -centralizer and has the formthen is proper if and only if the following conditions are satisfied:(1)There exists an -linear map such that(i) is a centralizer on and(ii) for all .(2) for all .

Proof. Since is a Lie -centralizer on , it follows that satisfies Theorem 6. Assume that the assumptions (1) and (2) hold, we define two maps satisfying and . Clearly, . We claim that . Indeed, according to (6), it suffices to prove that for all . Since is a centralizer and is a Lie -centralizer, it follows from the assumption (1)(ii) and Lemma 2 thatfor all . Therefore, . By Lemma 5, it remains to show that is a centralizer on .
According to Lemma 3 and the assumption (1)(i), it suffices to prove that satisfies the following conditions: is a centralizer; , , , and for all .
Since satisfies , it follows fromthat . Similarly, for all . Therefore,Next, we define an -linear map by for all . For each , we getOn the other hand, we haveAccording to the assumption (2) and (25), we obtain . Therefore, satisfies . That is, is a centralizer.
Using and the fact that is a centralizer, we arrive atfor all . Similarly, we get for all .
Applying Theorem 6 yields that for all . Hence, and for all . Therefore, is a centralizer. Finally,for all .
Conversely, suppose that is proper, then there exists a centralizer and an -linear map such that . In view of (6), we get , where is an -linear map satisfying for all . On the other hand, is a centralizer on and by Lemma 3, are centralizers. According to Lemma 1, we get for all .

Using Theorem 7 and Lemma 4, we can give the next main result, which provides the sufficient conditions for any Lie -centralizer on a trivial extension algebra to be proper.

Corollary 8. Assume that is a trivial extension algebra satisfying (3) and is a Lie -centralizer on with the formThen, is proper if the following conditions are satisfied:(1)Every Lie -centralizer on is proper;(2) for all ;(3)One of the following two conditions holds:(i) and is faithful as a right -module;(ii) and is faithful as a left -module.

Proof. Since is a Lie -centralizer on , it follows from Theorem 6 that is a Lie -centralizer on . According to the assumption (1), there exists an -linear map such that is a centralizer and vanishes on all -th commutators of . By Theorem 7, it is sufficient to show that satisfies for all . Using Lemma 4, if the assumption (3)(i) or (3)(ii) holds, then we have , which implies for all .

Applying Theorem 7 to triangular algebras, we can obtain the following result.

Corollary 9. Let be a Lie -centralizer on a triangular algebra , then has the formwhere , and is proper if and only if there exists a linear map , satisfying the following conditions:(1) is a centralizer on ;(2) for all and .

Proof. In view of Theorem 6, we havefor all . That is,where . According to Theorem 7, the remaining part is true.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed to the study conception and design.

Acknowledgments

This study was supported by the Jilin Science and Technology Department (no. YDZJ202201ZYTS622) and the project of Jilin Education Department (no. JJKH20220422KJ).