Biderivations of triangular algebras

• Published in: Linear algebra and its applications Vol. 431; no. 9; pp. 1587 - 1602
• Main Author: BENKOVIC, Dominik
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.10.2009; Elsevier
• Subjects: Algebra; Associative rings and algebras; Biderivation; Derivation; Exact sciences and technology; Inner biderivation; Inner derivation; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Nest algebra; Operator theory; Sciences and techniques of general use; Triangular algebra; Triangular matrix algebra; Nest algebra; 15A78; Triangular matrix algebra; Derivation; 16W25; Biderivation; Inner derivation; 47L35; Inner biderivation; Triangular algebra; Triangular matrix; Matrix algebra; Operator algebra; Upper triangular matrix; Ring; Block matrix
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2009.05.029

Let A be a triangular algebra. A bilinear map φ : A × A → A is called a biderivation if it is a derivation with respect to both arguments. In this paper we define the concept of an extremal biderivation, and prove that under certain conditions a biderivation of a triangular algebra A is a sum of an extremal and an inner biderivation. The main result is then applied to (block) upper triangular matrix algebras and nest algebras. We also consider the question when a derivation of a triangular algebra is an inner derivation.