Frobenius extensions about centralizer matrix algebras

For a nonempty set 𝐶 of matrices in 𝑀 𝑛 (𝑅), we define the centralizer algebra of 𝐶 by 𝑆 𝑛 (𝐶, 𝑅) := {𝑎 ∈ 𝑀 𝑛 (𝑅) | 𝑎𝑐 = 𝑐𝑎 for all 𝑐 ∈ 𝐶}.

In case 𝐶 = {𝑐}, we write 𝑆 𝑛 (𝑐, 𝑅) for 𝑆 𝑛 ({𝑐}, 𝑅).

The study of centralizer matrix algebras is one of important topics in the theory of matrices, which has significant applications not only in abstract studies, but also in applied problems (see [1,2,3,4]). If 𝐶 consists of nilpotent matrices and 𝑅 is an algebraically closed field, then the variety consisting of nilpotent matrices in 𝑆 𝑛 (𝐶, 𝑅) is of great interest in understanding properties of semisimple Lie algebras (see [5,6]). Typical examples of centralizer matrix algebras include centrosymmetric matrix algebras (see [7,8]), the Auslander algebras of the truncated polynomial algebras (see [9]) and centralizer of a matrix over an algebraically closed field. It is commonly known that centrosymmetric matrices have significant applications in Markov processes (see [7]), engineering problems and quantum physics (see [10]).

An algebra extension 𝐴/𝐵 is called a Frobenius extension provided that 𝐴 is finitely generated projective as a right 𝐵-module and isomorphic to Hom( 𝐴 𝐵 , 𝐵 𝐵 ) as a 𝐵-𝐴-bimodule. They play a crucial role in various mathematical domains. These include representation theory (see [11,8]), knot theory, and solutions to the Yang-Baxter equation (see [12]). They also have significant applications in topological quantum field theories and the theory of codes (see [13,14]).

An interesting example of Frobenius extensions related to centralizer matrix algebras, due to Xi and Zhang (see [9]), is that the extension 𝑀 𝑛 (𝑅)/𝑆 𝑛 (𝑐, 𝑅) is always a separable Frobenius extension for an arbitrary field 𝑅 and 𝑐 ∈ 𝑀 𝑛 (𝑅). Another interesting example of Frobenius extensions about matrix algebras is that 𝑀 𝑛 (𝑅)/𝑅 is always a Frobenius extension for any algebra 𝑅 (see [15]). However, it remains unknown if 𝑆 𝑛 (𝑐, 𝑅)/𝑅 is a Frobenius extension. Here, the general question reads as follows.

Is 𝑆 𝑛 (𝑐, 𝑅)/𝑅 always a Frobenius extension?

𝑀 𝑛 (𝑅) 𝑆 𝑛 (𝑐, 𝑅) 𝑅 ?

The main purpose of this note is to provide an answer to the above question. To state our results precisely, we first introduce some notation and definitions.

Recall that a Jordan block, denoted by 𝐽 𝑛 (𝜆), is an 𝑛 × 𝑛 matrix with its diagonal filled with the eigenvalue 𝜆 and ones on the superdiagonal, and zeros elsewhere. Let 𝐽 𝑛 denote the 𝑛 × 𝑛 Jordan block with eigenvalue zero. A matrix 𝑐 is Jordan-similar to 𝑎 if 𝑐 = 𝑢 -1 𝑎𝑢 for some invertible matrix 𝑢, where 𝑎 is in Jordan canonical form. This form is characterized by having Jordan blocks along the diagonal, with each block corresponding to an eigenvalue of 𝑐.

Since every unitary ring 𝑅 is an algebra over the ring of integers Z, we adopt the term ‘algebra’ instead of ‘ring’ throughout this paper. Our main result reads as follows. For a field 𝑅, an 𝑅-algebra 𝐴 is a (separable) Frobenius 𝑅-algebra if 𝐴/𝑅 is a (separable) Frobenius extension. Additionally, we denote by char(𝑅) the characteristic of the field 𝑅 and 𝑅 the algebraic closure of 𝑅. Inspired by the method of Xi and Zhang in [16], we obtain the following corollary. This article is outlined as follows. Section 2 is dedicated to some foundational concepts of Frobenius extensions. Section 3 presents the proof of Theorem 1.1. In addition, Section 3 also gives corollaries and examples which contribute to showing the main result.

In this section we discuss basic properties of separable Frobenius extensions and centralizer matrix algebras. Throughout the paper, 𝑅 denotes an algebra over an integral domain 𝕜. In addition, we denote by 𝑀 𝑚×𝑛 (𝑅) the set of all 𝑚 × 𝑛 matrices over 𝑅 and by 𝑒 𝑖, 𝑗 the matrix units of 𝑀 𝑚×𝑛 (𝑅) with 𝑖 ∈ [𝑚], 𝑗 ∈ [𝑛]. We write 𝐼 𝑛 for the identity matrix in 𝑀 𝑛 (𝑅). For a matrix 𝑐 ∈ 𝑀 𝑚×𝑛 (𝑅), we denote by 𝑡𝑟 (𝑐) the trace of 𝑐.

First, let us recall some facts about separable Frobenius extensions.

Lemma 2.1. [18] The following are equivalent:

Similarly, we have

On the other hand, assume 𝐵 1 𝐵 2 /𝐴 is a Frobenius extension. Let (𝐸, (𝑋 1 𝑖 , 𝑋 2 𝑖 ), (𝑌 1 𝑖 , 𝑌 2 𝑖 )) be a Frobenius system of (𝐵 1 𝐵 2 )/𝐴. Let 𝐸 1 (𝑏 1 ) = 𝐸 (𝑏 1 , 0) and 𝐸 2 (𝑏 2 ) = 𝐸 (0, 𝑏 2 ), for 𝑏 1 ∈ 𝐵 1 and 𝑏 2 ∈ 𝐵 2 . Then it is easy to check (𝐸 1 , 𝑋 1 𝑖 , 𝑌 1 𝑖 ) and (𝐸 2 , 𝑋 2 𝑖 , 𝑌 2 𝑖 ) are Frobenius systems. For the separable Frobenius extension, assume there exist 𝑑 1 ∈ 𝐵 1 and 𝑑 2 ∈ 𝐵 2 such that

Similarly, there exists

□ The direct sum of matrices is defined as follow.

Definition 2.5. Given an 𝑚 × 𝑛 matrix 𝑎 and a 𝑝 × 𝑞 matrix 𝑏, their direct sum, denoted by 𝑎 𝑏, is defined as a (𝑚 + 𝑝) × (𝑛 + 𝑞) matrix:

where the zero matrices have appropriate sizes to fill the blocks, thereby making 𝑎 and 𝑏 the diagonal blocks of the resulting matrix.

Secondly, let us