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.
Keywords: triangular automorphism, commutative ring, triangular derivation, filtered module, filtered algebra, graded algebra, triangular linear map, unitriangular group, polynomial ring, differential polynomial
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