39 research outputs found
Domestic canonical algebras and simple Lie algebras
For each simply-laced Dynkin graph we realize the simple complex Lie
algebra of type as a quotient algebra of the complex degenerate
composition Lie algebra of a domestic canonical algebra
of type by some ideal of that is
defined via the Hall algebra of , and give an explicit form of .
Moreover, we show that each root space of has a basis
given by the coset of an indecomposable -module with root easily
computed by the dimension vector of .Comment: 43 pages, 5 figures, revised versio
Relative Koszul coresolutions and relative Betti numbers
Let be a generator and a cogenerator in the category of finitely
generated right -modules for a finite-dimensional algebra over a filed
, and the additive closure of . We will define a
-relative Koszul coresolution of an
indecomposable direct summand of , and show that for a finitely
generated -module , the -relative -th Betti number for
at is given as the -dimension of the -th homology of the
-relative Koszul complex
of at for all . This is applied to investigate the minimal
interval resolution/coresolution of a persistence module , e.g., to check
the interval decomposability of , and to compute the interval approximation
of .Comment: 21 page