Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical -Matrices

Abstract

Our aim is to classify the Rota-Baxter operators of weight 0 on the 3-dimensional Lie algebra whose derived algebra’s dimension is 2. We explicitly determine all Rota-Baxter operators (of weight zero) on the 3-dimensional Lie algebras . Furthermore, we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras and the induced left-symmetry algebra structures on .

1. Introduction

In physics, the Yang-Baxter equation is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that, in some scattering situations, particles may preserve their momentum while changing their quantum internal states. Rota-Baxter algebra started with the probability study and has since found applications in many areas of mathematics and physics, such as quasi-symmetric functions, number theory, dendriform algebras, and Yang-Baxter equations.

A Rota-Baxter operator (of weight zero) on an associative algebra is defined to be a linear map satisfying Rota-Baxter operators (on associative algebras) were introduced by Baxter to solve an analytic formula in probability [1–4]. It has been related to other areas in mathematics and mathematical physics [5–9]. A Rota-Baxter operator (of weight zero) on a Lie algebra is a linear operator such that

In fact, a Rota-Baxter operator is also called the operator form of the classical Yang-Baxter equation [10–13]. Let be a Lie algebra and . is called a classical -matrix if it is a solution of the classical Yang-Baxter equation (CYBE) in : that is,in , where is the universal enveloping algebra of and Set . It is easy to obtain that is skew-symmetric if and only if . Semenov-Tian-Shansky proved in [14] that is skew-symmetric and there is a nondegenerate symmetric invariant bilinear form on Lie algebra ; relation (2) is equivalent to relation (3) when the weight is zero. Furthermore, Rota-Baxter operators of weights and on a Lie algebra give rise to solutions of CYBE on the double Lie algebra over the direct sum of the Lie algebra and its dual space [12, 15, 16]. Moreover, we can get some solutions of CYBE in Lie algebras through Rota-Baxter operators of any weight on .

In [12], the authors gave all Rota-Baxter operators (of weight zero) on 3-dimensional simple Lie algebra . The aim of this paper is to determine the Rota-Baxter operators (of weight zero) on the 3-dimensional Lie algebra which is not simple, and the dimension of its derived algebra is 2. We will determine the Rota-Baxter operators on the Lie algebra and give a family of solutions of CYBE in . This paper is organized as follows. In Section 2, we give the classification theorem of Rota-Baxter operators (of weight zero) on . In Section 3, we give the corresponding solutions of CYBE in . In Section 4, we give the corresponding left-symmetry structure on .

2. The Rota-Baxter Operators on (of Weight Zero)

2.1. Notations and the Classification Theorem

Let be a 3-dimensional linear Lie algebra whose standard (Cartan-Weyl) basis consists of , , over the field of complex numbers with the following Lie brackets: Thus, a linear operator is determined by where . is a Rota-Baxter operator on if the above matrix satisfies (2). Here is our main theorem.

Theorem 1. All Rota-Baxter operators of weight zero on are listed in their matrices form with respect to the Cartan-Weyl basis below, where , , and are nonzero complex numbers.

2.2. Reduction to Quadratic Equations

In order to show that is a Rota-Baxter operator, we only check the following: It follows from (5) and (6) that while Comparing the coefficients in (9) and (10), we have Similarly, from we obtain the following six equations:

2.3. Solving the Quadratic Equations

Equation (19) implies . To solve the quadratic equations (11), (12), (13), (15), (16), (17), (18), (19), and (20), we distinguish the following cases depending on whether or not.

Case 1. , . That is, , , taking . Equation (16) implies . Equation (15) implies . Equation (11) implies . Equation (12) implies . Equation (13) implies . We obtain Taking , , we obtain . Taking , , we obtain . Taking , , we obtain . Taking , , we obtain .

Case 2. Assume , . That is, , . We distinguish the two cases depending on whether or not.
Subcase 2.1. If , then (13) implies . Equation (15) implies . Equation (11) implies .
Subcase 2.1.1. If , , we obtain Taking , , we obtain . Taking , , we obtain . Taking , , we obtain . Taking , , we obtain .
Subcase 2.1.2. If , , taking , we obtain Taking , , we obtain . Taking , , we obtain . Taking , , we obtain . Taking , , we obtain .
Subcase 2.1.3. If , , taking , we obtain Taking , , we obtain . Taking , , we obtain . Taking , , we obtain . Taking , , we obtain .
Subcase 2.2. If , taking , (17) implies . Then (15) implies . Equation (13) implies . That is, .
Subcase 2.2.1. If , , , and then , , we obtainTaking ,  , , we obtain . Taking , ,   , we obtain . Taking , ,   , we obtain . Taking , , , we obtain . Taking , , , we obtain . Taking , ,   , we obtain . Taking , , , we obtain . Taking , ,   , we obtain .
Subcase 2.2.2. If , , and then , , (3.4) implies . Then we have , , giving a contradiction.
Subcase 2.2.3. If , , that is, , , and then , , we obtainTaking , , we obtain . Taking ,  , we obtain . Taking , , we obtain . Taking , , we obtain .

Case 3. Assume ,  . Taking , then . Equation (16) implies . Equations (12) and (18) imply . Equation (17) implies . Equation (15) implies . Equations (11), (15), and (17) imply . Then we have . So , , , . We obtain Taking , , we obtain . Taking , , we obtain . Taking , , we obtain . Taking , , we obtain .

3. Solutions of the CYBE in

In this section, we will give some solutions of CYBE in . Let be a Lie algebra and a representation of . On the vector space , there is natural Lie algebra structure (denoted by given by Let be the dual representation of . A linear map can be identified as an element in as follows. Let be a basis of and the dual basis in : that is, . Let be a basis of . Set . Since, as a vector space, , then

Lemma 2 (see [15]). Let be a Lie algebra; let be a -module. A linear map is a Rota-Baxter operator if and only if is a skew-symmetric solution of CYBE in .

Now consider the adjoint representation of which is a -module. Let , , be the Cartan-Weyl basis. Using Lemma 2 and relation (29), we can obtain a family of solutions of CYBE in through the Rota-Baxter operators on given in Theorem 1.

Theorem 3. The following tensors are solutions of the classical Yang-Baxter equation in , where , , and are nonzero complex numbers

One can check that all of the tensors above are solutions of the classical Yang-Baxter equation in .

4. Induced Left-Symmetric Algebras from Rota-Baxter Operators of Weight 0 on

A left-symmetric algebra structure on is a bilinear product satisfying the conditionfor all . There are many examples of Lie algebras which do not admit a left-symmetric product. For example, it is easy to see that there are no left-symmetric algebras with semisimple Lie algebra. Equation (31) implies that the commutators satisfy the Jacobi identity; that is to say each left-symmetric product has an associated commutation Lie algebra, which is called the subadjacent Lie algebra. If is a Rota-Baxter operator on a left-symmetric algebra, then is a solution of CYBE on its subadjacent Lie algebra [17]. Clearly, each associative algebra product is a left-symmetric product. Given a Lie algebra , it is a fundamental problem to decide whether admits a left-symmetric product and to give a classification of such products [18]. As an application of Yang-Baxter operators, we can use them to construct left-symmetric algebras with respect to .

Lemma 4 (see [13]). Let be a Lie algebra; is called a solution of the classical Yang-Baxter equation. Define a new operation on by Then is a left-symmetric algebra.

According to Theorem 3 and Lemma 4, we can get some left-symmetric algebras of .

Theorem 5. Some left-symmetric algebras of (of weight zero) are determined:(1), , ;(2), , , , , ;(3), ;(4), , , , , ;(5);(6);(7);(8), ;(9), ;(10), ;(11);(12), , ;(13), , ;(14), , ;(15), ;(16), , ;(17), , ;(18), , ;(19), , ;(20), , ;(21), ;(22), ;(23), , ;(24), , ;(25), , ;(26), ;(27), , ;(28), , , , , , , , ;(29), , , , , ;(30), ;(31), , , , , , , , .

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work is supported by the National Science Foundation of China (Grants nos. 11047030 and 11771122) and the Science and Technology Program of Henan Province (Grant no. 152300410061).