Triangular Algebra
Suppose that A and B are two algebras and M is a unital A-B-bimodule. Then
![画像: [A M; 0 B]={[a m; 0 b]:a in A,m in M,b in B}](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TriangularAlgebra/NumberedEquation1.svg){kind=link}
with the usual 2×2 matrix-like addition and matrix-like multiplication is an algebra.
An algebra T is called a triangular algebra if there exist algebras A and B and an A-B-bimodule M such that T is (algebraically) isomorphic to
![画像: [A M; 0 B]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TriangularAlgebra/NumberedEquation2.svg){kind=link}
under matrix-like addition and matrix-like multiplication.
For example, the algebra T_n of n×n upper triangular matrices over the complex field C may be viewed as a triangular algebra when n>1.
This entry contributed by Mohammad Sal Moslehian
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References
Cheung, W.-S. "Commuting Maps of Triangular Algebras." J. London Math. Soc. 63, 117-127, 2001.Forrest, B. E. and Marcoux, B. E. "Derivations of Triangular Banach Algebras." Indiana Univ. Math. J. 45, 441-462, 1996.Referenced on Wolfram|Alpha
Triangular AlgebraCite this as:
Moslehian, Mohammad Sal. "Triangular Algebra." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TriangularAlgebra.html