Open AccessConnected cochain DG algebras of Calabi-Yau dimension 0
Author(s) -
Jiwen He,
Xuefeng Mao
Publication year - 2016
Publication title -
proceedings of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.968
H-Index - 84
eISSN - 1088-6826
pISSN - 0002-9939
DOI - 10.1090/proc/13081
Subject(s) - annotation , algorithm , type (biology) , computer science , semantics (computer science) , artificial intelligence , mathematics , biology , programming language , ecology
Let A A be a connected cochain differential graded (DG, for short) algebra. This note shows that A A is a 0 0 -Calabi-Yau DG algebra if and only if A A is a Koszul DG algebra and T o r A 0 ( k A , A k ) \mathrm {Tor}_A^0(\Bbbk _A,{}_A\Bbbk ) is a symmetric coalgebra. Let V V be a finite dimensional vector space and w w a potential in T ( V ) T(V) . Then the minimal subcoalgebra of T ( V ) T(V) containing w w is a symmetric coalgebra, which implies that a locally finite connected cochain DG algebra is 0 0 -CY if and only if it is defined by a potential w w .