Rota–Baxter operators on a sum of fields

Gubarev, Vsevolod

doi:10.1142/s0219498820501182

Cited by 7 publications

(16 citation statements)

References 24 publications

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“…In [14], RB-operators of nonzero weight on a sum of fields were studied. In particular, it was proved that Statement 7 [14].…”

Section: Rb-operators On Fmentioning

confidence: 99%

“…In [14], RB-operators of nonzero weight on a sum of fields were studied. In particular, it was proved that Statement 7 [14]. Let R be a not inner-splitting RB-operator of weight 1 on the sum of fields F 3 = F f 1 ⊕ F f 2 ⊕ F f 3 , then up to permutation of coordinates and action of φ, we have nine cases:…”

Section: Rb-operators On Fmentioning

confidence: 99%

“…If β = γ = 0, then R is splitting. If β = γ = 1, R is conjugate to the RB-operator 3-II) with the help of Φ 13 • T • ̺, where ̺ is defined by (14) with a = c = 1/b.…”

Section: )mentioning

confidence: 99%

“…Finally, ψ −1 P ψ is the RB-operator 6-V). Here ψ = Φ 23 ̺Φ 23 , where ̺ is defined by (14) with the parameters a = c = −1/b.…”

Section: )mentioning

confidence: 99%

“…Conjugation of the RB-operator 5-IV) with the automorphism Φ 23 ̺Φ 23 coincides with the RB-operator from the case 2-I). Here ̺ is defined by (14) with a = c = 1/b. Also, conjugation of the RB-operator 2-II) with the same ̺ gives the RB-operator 5-V).…”

Section: )mentioning

confidence: 99%

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

Goncharov

Gubarev

2020

Linear and Multilinear Algebra

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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 − .

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“…In [14], RB-operators of nonzero weight on a sum of fields were studied. In particular, it was proved that Statement 7 [14].…”

Section: Rb-operators On Fmentioning

confidence: 99%

Section: Rb-operators On Fmentioning

confidence: 99%

“…If β = γ = 0, then R is splitting. If β = γ = 1, R is conjugate to the RB-operator 3-II) with the help of Φ 13 • T • ̺, where ̺ is defined by (14) with a = c = 1/b.…”

Section: )mentioning

confidence: 99%

“…Finally, ψ −1 P ψ is the RB-operator 6-V). Here ψ = Φ 23 ̺Φ 23 , where ̺ is defined by (14) with the parameters a = c = −1/b.…”

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

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