Rota–Baxter operators of nonzero weight on the matrix algebra of order three

Abstract

We classify all Rota Baxter operators of nonzero weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.Keywords: Rota Baxter operator, matrix algebra. RB-operators on M n (F )Over a field F , denote the algebra of all upper and lower triangular matrices of order n by U n and L n respectively.Example 1 [7]. Decomposing M n (F ) = L n ⊕ D n ⊕ U n (as vector spaces) and given an RB-operator defined on D n , we have a triangular-splitting RB-operator defined withDecomposing M n (F ) = M 1 ⊕M 2 ⊕M 3 (as vector spaces) and given an RB-operator defined on M 1 , we have a triangular-splitting RB-operator on M n (F ) defined with A + = M 2 ,Example 3. Let an algebra A be equal a direct vector space sum A − ⊕ A 0 ⊕ A + of its three subalgebras satisfying the conditions A 0 A − , A − A 0 ⊂ A + . Suppose that P is an RBoperator of weight 1 on A 0 . Then a linear operator R defined on A as R(a − + a 0 + a + ) = P (a 0 ) − a + is an RB-operator of weight 1 if R(A 0 )A − , A − R(A 0 ) ⊂ A − .

Deformations of associative Rota-Baxter operators

“…The identity (15) holds for n = 0 as T 0 = T is a λ-weighted relative Rota-Baxter operator. For n = 1, we get…”

“…They are related to post-algebras [5], weighted infinitesimal bialgebras [27], weighted associative Yang-Baxter equations [27], combinatorics of planar rooted forests [26], and play an important role in mathematical physics [5]. Some classification result of weighted Rota-Baxter operators on matrix algebras are given in [15]. See [4,11] for some other generalizations of Rota-Baxter operators.…”

Spectrum of Rota-Baxter operators

“…. , s − 2, s − 1} with r + s = n. By Lemma 1 from [13], all subspaces V q = Span{e ij | i − j = q} are R-invariant. Since all V q for q = 0 have dimension less or equal to n − 1, we have (R(R − id)) n−1 = 0 on each of such V q .…”

“…Let us generalize the RB-operator (M1) on M 2 (F ) from [16,Thm. 4.13] and the RB-operator 1-I on M 3 (F ) from [13,Thm. 3].…”

“…There is a plenty of papers devoted to a partial or complete classification of Rota Baxter operators on a given small-dimensional algebra: sl 2 (F ) [22,28,29], sl 3 (F ) [23], M 2 (F ) [2,8,34], M 3 (F ) [13,33], the Grassmann algebra Gr 2 [8], and others, see the survey [16] and the references therein.…”