Classification of Rota-Baxter operators on semigroup algebras of order two and three

Abstract In this paper, we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to predict and verify these solutions.


Introduction
Rota-Baxter operators arose from the probability study of G. Baxter in 1960 [7], defined by the operator identity P x ð ÞP y ð Þ ¼ P xP y ð Þ À Á þ P P x ð Þy ð Þþ kP xy ð Þ ; where k is a fixed scalar called the weight. When k ¼ 0, the operator is a natural algebraic generalization of the integral operator. In the 1960s and 70 s, these operators attracted attention from well-known analysts such as Atkinson [2] and combinatorialists such as Cartier and Rota [10,39,40]. In the 1980s these operators were studied in integrable systems as the operator form of the classical Yang-Baxter equations [41], named after the well-known physicists C.N. Yang and R.J. Baxter. Since the late 1990s, the study of Rota-Baxter operators has made great progress both in theory and in applications in combinatorics, number theory, operads, boundary value problems and mathematical physics [3-6, 9, 11, 17, 20-22, 38]. Rota-Baxter algebras arising naturally from applications as well as theoretical investigations (e.g. free Rota-Baxter algebras [16,21]) are mostly infinite dimensional. To study finite dimensional Rota-Baxter algebras in general, it is useful to start with low dimensional Rota-Baxter algebras. Even there computations are quite complicated. In recent years, some progress regarding such computations has been achieved, with applications to pre-Lie algebras, dendriform algebras and the classical Yang-Baxter equation [1,28,35,43]. In this paper, we study Rota-Baxter operators on a class of low dimensional algebras, namely semigroup algebras for small order semigroups.
Semigroup algebras, a natural generalization of group algebras, form an important class of associative algebras arising from semigroups [34]. The representation of semigroups leads to a semigroup algebra satisfying polynomial identities. In this regards, Rota-Baxter operators on a semigroup algebra can be regarded as an operated semigroup algebra satisfying an operator identity. It has been shown that every finite dimensional algebra of finite representation type over an algebraically closed field is a contracted semigroup algebra. Recently, semigroup algebras have experienced rapid development on the theoretical side [13,14,24,25] as well as in applications to representation theory, cohomology, geometric group theory, topology, combinatorics, algebraic geometry, and number theory [8,19,26,27,[30][31][32][33]42]. Thus studying semigroup algebras in their canonical basis has a significance of its own.
In this paper, we classify all Rota-Baxter operators on semigroup algebras of order 2 and 3. Through studying Rota-Baxter operators on low dimensional semigroup algebras, we hope to find patterns for the study of Rota-Baxter operators on general semigroup algebras. See Section 6 for further details. Rota-Baxter operators on associative algebras of dimension 2 and 3 have been determined up to isomorphism in [28]. Here we focus on the particular presentation of such an algebra in terms of the canonical semigroup algebra basis, because of the aforementioned importance of using such a basis. Indeed, as one notices by comparing the classifications given here with the ones in [28], the resulting Rota-Baxter operators take a very different form.
Because of the complex nature of Rota-Baxter operators, determining their classification by hand is challenging even for low dimensional algebras, as observed in [1,28,43]. In such a case, computer algebra provides an indispensable aid for both predicting and verifying these operators. Nevertheless, for ensuring theoretical accuracy, it is still necessary to carry out a rigorous proof of the classification. In Section 2, we start by developing the general setup of the equations that serve as the necessary and sufficient conditions characterizing a Rota-Baxter operator on a semigroup algebra. We then provide the Mathematica procedure that has helped us in solving the classification problem. In Section 3, we classify all Rota-Baxter operators on semigroup algebras of order 2. For Rota-Baxter operators on semigroup algebras of order 3, we carry out the classification in two sections, with Section 4 for commutative semigroup algebras and Section 5 for noncommutative semigroup algebras. We end the paper with some conclusion remarks.

The general setup and the computer algebra procedure
In this section, we first formulate the general setup for determining Rota-Baxter operators of weight zero on a semigroup algebra. We then implement this setup in Mathematica to obtain a procedure that helped us to obtain classifications of Rota-Baxter operators on semigroup algebras of order two and three.

The general setup
In this subsection, we give the general setup of Rota-Baxter operators on a semigroup algebra in matrix form. Let S be a finite semigroup with multiplication Á that we often suppress. Thus S ¼ fe 1 ; :::; e n g. Let k be a commutative unitary ring and let a m e m ja m 2 k; 16m6n denote the semigroup algebra of S. The order n of the semigroup S is also said to be the order of the semigroup algebra k½S.
Let P : k½S ! k½S be a Rota-Baxter operator of weight zero. Since P is k-linear, we have The matrix C :¼ C P :¼ ðc ij Þ 16i;j6n is called the matrix of P. Further, P is a Rota-Baxter operator if and only if Let the Cayley (multiplication) table of the semigroup S be given by where r m k' 2 f0; 1g. Then we have r m k' c ik c j' e m and P P e i ð Þ e j þ e i P e j ð Þ Thus we obtain Theorem 2.1. Let S ¼ fe 1 ; :::; e n g be a semigroup with its Cayley table given by Eq. (5). Let k be a commutative unitary ring and let P : k½S ! k½S be a linear operator with matrix C :¼ C P ¼ ðc ij Þ 16i;j6n . Then P is a Rota-Baxter operator of weight zero on k½S if and only if the  following equations hold.
We will determine the matrices C P for all Rota-Baxter operators P on k½S of order two or three.

The Mathematica procedure
In this subsection, we describe the computer algebra procedure (implemented in Mathematica) for computing the Rota-Baxter operators on semigroup algebras of semigroup of order 3, listed in Tables 1 and 2. This procedure serves both for guiding and verifying the manual proofs of the classification theorems carried out in later sections of the paper.
The Mathematica code and accompanying syntax definitions are given in Figure 1. The function RBA with four arguments creates Eq. (6) for a fixed pair of elements, which are then instantiated by all generator pairs. For added clarity, we have also displayed the general form of these equations for the generic 2 Â 2 Cayley table defined at the beginning. The main function for determining Rota-Baxter operators is FindRBO, which works by solving the equations created by RBA. For converting a given Cayley table to the structure constants r m kl used in Eq. (6), the function SGM is employed.
We illustrate these functions by considering the first semigroup t ¼ CSð1Þ of Table 1; for the detailed computation we refer to Section 4.2.1. The underlying set fe 1 ; e 2 ; e 3 g of CS(1) here will be simplified to {1, 2, 3}. The Mathematica code above yields the results given in Figure 2. In fact, the output gives two Rota-Baxter operators for CS(1), but the second is a special case of the first. Let pð1; 1Þ ¼ a; pð2; 1Þ ¼ b; pð1; 2Þ ¼ c; pð2; 2Þ ¼ d; pð1; 3Þ ¼ e; pð2; 3Þ ¼ f , where a; b; c; d; e; f 2 k. Then pð3; 1Þ ¼ ÀaÀb; pð3; 2Þ ¼ ÀcÀd and pð3; 3Þ ¼ ÀeÀf , so we obtain the matrix the transpose of which is given in Table 3 as C 1;1 .

Rota-Baxter operators on semigroup algebras of order 2
In this section, we determine all Rota-Baxter operators on semigroup algebras k½S of order 2.
As is well known [37], there are exactly five distinct nonisomorphic semigroups of order 2. We use N 2 ; L 2 ; R 2 ; Y 2 and Z 2 respectively to denote the null semigroup of order 2, the left zero semigroup, right zero semigroup, the semilattice of order 2 and the cyclic group of order 2. Since  L 2 and R 2 are anti-isomorphic, there are exactly four distinct semigroups of order 2, up to isomorphism and anti-isomorphism, namely N 2 , Y 2 , Z 2 and L 2 . Let fe 1 ; e 2 g denote the underlying set of each semigroup. Then the Cayley tables for these semigroups are as follows: Theorem 3.1. Let k be a field of characteristics zero. All Rota-Baxter operators on a semigroup algebra k½S of order 2 have their matrices C P given in Table 4, where all the parameters are in k and RBO ðresp. SAÞ is the abbreviation of Rota-Baxter operator ðresp. semigroup algebraÞ. Proof. We divide the proof of the theorem into four cases, one for each of the four semigroups S in Table 5. For each case, by Theorem 2.1, P is a RBO on k½S if and only if the eight Equations (6) hold (with 16i; j; m62). So we just need to solve these equations. It is straightforward to verify that what we obtain does satisfy all equations. Let 0 2Â2 denote the 2 Â 2 zero matrix. Case 1. Let S ¼ N 2 : In Eq. (6), taking i ¼ j ¼ 1 with 16m62 and i ¼ j ¼ 2 with 16m62, we get  Table 3. RBOs on commutative semigroup algebras of order 3.

CS of order 3
Matrices of RBOs on semigroup algebras CS of order 3 Matrices of RBOs on semigroup algebras Assume c 11 þ c 12 6 ¼ 0. Then by Eqs. (7) and (8) It is straightforward to check that they also satisfy the other equations in Eq. (6). Hence these are all the matrices C P for Rota-Baxter operators on k½S.
From Eq. (11) By Eqs. (19) and (22), we have c 2 11 ¼ c 2 22 . By Eqs. (20) and (21) On the other hand, This completes the proof of Theorem 3.1. Up to isomorphism and anti-isomorphism, there are 18 semigroups of order 3 [12,15,18]. The Cayley tables of the 18 semigroups of order 3 can be found in [18]. See also [12,29,36]. We denote by CS and NCS the class of 12 commutative semigroups and the class of 6 noncommutative semigroups, respectively. When a semigroup S has order 3, the equations in Eq. (6) for the matrix C P of a Rota-Baxter operator P on k½S are given by the following 27 equations.
In this section, we determine the Rota-Baxter operators on the semigroup algebras for the 12 commutative semigroups of order 3 in Table 1. Rota-Baxter operators on semigroup algebras for the 6 noncommutative semigroups of order 3 will be determined in Section 5.

Statement of the classification theorem in the commutative case
A classification of the 12 commutative semigroups of order 3 is given in Table 1.
We have the following classification of Rota-Baxter operators on order 3 commutative semigroup algebras. The proof will be given in Section 4.2.
Theorem 4.1. Let k be a field of characteristic zero. The matrices of Rota-Baxter operators on 3dimensional commutative semigroup algebras are given in Table 3, where all the parameters take values in k and RBO ðresp. CSÞ is the abbreviation for Rota-Baxter operator ðresp. commutative semigroupÞ.

Proof of Theorem 4.1
We will prove Theorem 4.1 by considering each of the 12 commutative semigroups CSðiÞ; 16i612; of order 3 in Table 1. For each semigroup, we solve some of the equations in Eq. (23) for the Cayley table of the corresponding semigroup. It is straightforward to verify that what we obtain this way indeed satisfies all the equations in Eq. (23). Let 0 3Â3 denote the 3 Â 3 zero matrix.

The proof for k½CSð1Þ
We prove that the matrices C P ¼ ðc ij Þ 16i;j63 of all the Rota-Baxter operators P on the semigroup algebra k½CSð1Þ are given by C 1;1 in Table 3.
Applying the Cayley table of CS(1) in Eq. (23) and taking Assume c 11 þ c 12 þ c 13 6 ¼ 0: Then by Eqs. (24), (25) and (26) Since they can be checked to satisfy other equations in Eq. (23), they give the matrices of all the Rota-Baxter operators on k½CSð1Þ.

The proof for k½CSð2Þ
Here we prove that the matrices C P ¼ ðc ij Þ 16i;j63 of all the Rota-Baxter operators P on the semigroup algebra k½CSð2Þ are given by C 2;1 and C 2;2 in Table 3.

The proof for k½CSð3Þ
Here we prove that the matrices C P ¼ ðc ij Þ 16i;j63 of all the Rota-Baxter operators P on the semigroup algebra k½CSð3Þ are given by C 3;1 in Table 3.
Applying the Cayley table of CS(3) in Eq. (23) and taking i ¼ It can be checked that they also satisfy the other equations in Eq. (23) and hence give all the Rota-Baxter operators on k½CSð3Þ.

The proof for k½CSð4Þ
We prove that the matrices C P ¼ ðc ij Þ 16i;j63 of all the Rota-Baxter operators P on the semigroup algebra k½CSð4Þ are given by C 4;1 in Table 3.
Applying the Cayley table of CS(4) in Eq. (23) and taking i ¼ j ¼ 1 with 16m63; i ¼ j ¼ 2 with 16m63 and i ¼ j ¼ 3 with 16m63, we obtain The proofs for the semigroups CSð6Þ; CSð8Þ, and CS(10) are similar to the proof for CS (4). So their proofs are omitted here but could be found in the on-line version [23]. Likewise the proofs for the semigroups CSð5Þ; CSð7Þ; CSð9Þ, and CS(11) are similar to the proof for CS(3) and hence is left in [23].

The proof for k½CSð12Þ
We finally prove that the matrices C P ¼ ðc ij Þ 16i;j63 of the Rota-Baxter operators P on the semigroup algebra k½CSð12Þ are given by C 12;1 in Table 3.

Semigroups
Matrices of Rota-Baxter operators on semigroup algebras NCSð1Þ

Rota-Baxter operators on noncommutative semigroup algebras of order 3
In this Section, we classify all Rota-Baxter operators on noncommutative semigroup algebras of order 3.

Statement of the classification theorem in the noncommutative case
A classification of the six noncommutative semigroups of order 3, up to isomorphism and antiisomorphism, is given in Table 2.
In summary, we get get solutions of Eqs. (77)-(91) It can be checked that they also satisfy the other equations in Eq. (23) and hence give matrices of Rota-Baxter operators on k½NCSð1Þ. 5.2.2. The proof for k½NCSðiÞ where i ¼ 2; 3; 4; 6 The proof of these cases are similar to the one for k½NCSð1Þ in that the proofs are carried out by (iterated) bisecting depending on whether or not certain elements are zero. Details of the proofs are provided in [23]. So we next move on to the proof of k½NCSð5Þ.

The proof for k½NCSð5Þ
We next prove that the matrices of the Rota-Baxter operators on the semigroup algebra k½NCSð5Þ are given by N 5;i ; 16i622; in Table 6.
Applying the Cayley table of NCS(5) in Eq. (23) and then taking i ¼ j ¼ 1 with 16m63; i ¼ 2; j ¼ 1 with 16m63 and i ¼ 3; j ¼ 1 with 16m63, we obtain We divide the proof into five cases.
With the remark made in Section 5.2.2, now the proof of Theorem 5.1 is completed.

Conclusion
We have presented a complete and explicit classification of Rota-Baxter operators on semigroup algebras for the orders 2 and 3. With some care taken to ensure efficient calculations, the same approach could be used for classifying all semigroup algebras over semigroups of order 4. This would provide a valuable stock of "finite" exemplary objects in the Rota-Baxter category. It would also be interesting to explore Rota-Baxter structures on other classes of algebras, e.g. low-dimensional path algebras, matrix rings, and special types of group algebras. As an example for the latter, consider cyclic groups: By the results presented above we know that all Rota-Baxter operators over the cyclic group of order 2 or 3 are trivial-is this true for any (prime order) cyclic group?