CM-自由代数的投射模范畴的实现问题及应用

If the projective module categories of two algebras are equivalent, then their module categories are equivalent. However, if the Gorenstein projective module categories of two algebras are equivalent, then their module categories may be not equivalent. So there is a natural question: given a category, how can we find all algebras such that their Gorenstein projective module categories are equivalent to the given category? This is a realization problem of categories. In this project we assume the given category is the projective module category of a CM-free algebra, and try to solve the above realization problem. On the other hand, since algebras with finite global dimension, especially the quasi-hereditary algebras are CM-free algebras, we want to find applications of the realization problem on quasi-hereditary algebras and Lie algebras.

若两个代数的投射模范畴等价，则它们的表示范畴等价。但是，若两个代数的Gorenstein投射模范畴等价，则它们的表示范畴不一定等价。故有一个自然的问题： 给定一个范畴，如何找到所有的代数，使其Gorenstein投射模范畴等价于这个范畴？ 这是范畴的一个实现问题。 本项目假定给定的是一个CM-自由代数的投射模范畴，来寻求它的实现。另外，由于CM-自由代数囊括了所有的 整体维数有限的代数，特别是拟遗传代数，所以我们希望能找到上述实现问题在拟遗传代数和李代数上的应用。