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Cluster‐tilted algebras as trivial extensions
Author(s) -
Assem I.,
Brüstle T.,
Schiffler R.
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm107
Subject(s) - mathematics , cluster (spacecraft) , pure mathematics , algebra over a field , computer science , programming language
Given a finite‐dimensional algebra C (over an algebraically closed field) of global dimension at most two, we define its relation‐extension algebra to be the trivial extension C ⋉ Ext C 2 ( D C , C )of C by the C – C ‐bimoduleExt C 2 ( D C , C ) . We give a construction for the quiver of the relation‐extension algebra in case the quiver of C has no oriented cycles. Our main result says that an algebraC ~is cluster‐tilted if and only if there exists a tilted algebra C such thatC ~is isomorphic to the relation‐extension of C .