
Contents
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Filtered modules Filtered modules
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Filtered algebras Filtered algebras
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Let A be a complete filtered K-algebra with respect to the filtration (Ai). Then every sequence (an) in A with an 2 An for each n is summable in A. In particular, given Pa formal power series f = n fnzn 2 K[[z]] (fn 2 K Pfor each n) and a 2 A1, the sequence (fnan) is summable in A, and we set f (a) := fnan. Here is an analogue of Lemma 12.1.1: Let A be a complete filtered K-algebra with respect to the filtration (Ai). Then every sequence (an) in A with an 2 An for each n is summable in A. In particular, given Pa formal power series f = n fnzn 2 K[[z]] (fn 2 K Pfor each n) and a 2 A1, the sequence (fnan) is summable in A, and we set f (a) := fnan. Here is an analogue of Lemma 12.1.1:
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Derivations Derivations
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Graded algebras Graded algebras
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Gradings of polynomial algebras Gradings of polynomial algebras
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12.2 Triangular Linear Maps 12.2 Triangular Linear Maps
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Locally nilpotent and locally unipotent endomorphisms Locally nilpotent and locally unipotent endomorphisms
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Diagonals Diagonals
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Notes and comments Notes and comments
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12.4 Derivations on the Ring of Column-Finite Matrices 12.4 Derivations on the Ring of Column-Finite Matrices
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12.5 Iteration Matrices 12.5 Iteration Matrices
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Bell polynomials Bell polynomials
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The iterative logarithm The iterative logarithm
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12.6 Riordan Matrices 12.6 Riordan Matrices
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The Riordan group The Riordan group
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The Lie algebra of R The Lie algebra of R
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Notes and comments Notes and comments
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12.7 DERIVATIONS ON POLYNOMIAL RINGS 12.7 DERIVATIONS ON POLYNOMIAL RINGS
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Triangular derivations Triangular derivations
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Diagonals Diagonals
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The Stirling automorphism The Stirling automorphism
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By the recurrence relation (5.7.3) for Stirling numbers of the first kind, (12.7.4) M0= –MD where D = diag(0; 1; 2; 3; : : : ). By the recurrence relation (5.7.3) for Stirling numbers of the first kind, (12.7.4) M0= –MD where D = diag(0; 1; 2; 3; : : : ).
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We also define the diagonal K-automorphism ϕ of A by We also define the diagonal K-automorphism ϕ of A by
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Twelve Triangular Automorphisms
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Published:June 2017
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Abstract
This chapter focuses on triangular automorphisms, which can be analyzed by Lie techniques. Throughout the discussion K is a commutative ring containing ℚ as a subring. A formalism is introduced to analyze triangular automorphisms of such a polynomial algebra by means of their logarithms, the triangular derivations. After presenting some definitions and simple facts about filtered modules, filtered algebras, and graded algebras, the chapter considers triangular linear maps and the Lie algebra of an algebraic unitriangular group. It then describes derivations on the ring of column-finite matrices, along with iteration matrices and Riordan matrices. It also explains derivations on polynomial rings and concludes by applying triangular automorphisms to differential polynomials.
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