Abstract

S. Ariki : Rouquier's theory of quasi-hereditary covers

In this short course, I explain Rouquier's use of integral highest weight categories and his deformation principle in finding the quasi-hereditary covers of Hecke algebras.

R. Kase : Quasi-hereditary algebras and highest weight categories

Quasi-hereditary algebras were introduced by Cline, Parshall and Scott in order to realize highest weight categories as module categories. In this talk, we recall definition of quasi-hereditary algebras and explain Cline-Parshall-Scott’s equivalence. Also, we refer to Iyama’s result for finiteness of representation dimension of finite dimensional algebras.

R. Kanda : Auslander-Buchweitz approximations and cotilting modules

I will give an introduction to Auslander-Buchweitz approximations based on [AB 1989], together with the connection to cotilting modules [AR 1991]. An application to quasi-hereditary algebras will be demonstrated following [Ringel 1991].

The papers [AB 1989] and [AR 1991] relate Cohen-Macaulay representation theory and tilting theory by making an analogy between canonical modules and cotilting modules. I will explain these theories from the viewpoint of cotorsion pairs in abelian categories.

[AB 1989] shows that for a commutative noetherian Cohen-Macaulay local ring with a canonical module, every module over the ring has two kinds of approximations into maximal Cohen-Macaulay modules and modules of finite injective dimension. [AB 1989] gives more general statements using abelian categories, which can also be applied to Iwanaga-Gorenstein rings.

The paper [AR 1991] illustrates how to construct a cotorsion pair from a cotilting module, in the sense of Miyashita and Happel, over an artin algebra. We will also see a correspondence between basic cotilting modules and contravariantly-finite resolving subcategories and covariantly-finite coresolving subcategories, which becomes bijective if the algebra has finite global dimension.

For a quasi-hereditary algebra, the subcategories arising from standard modules and costandard modules form a cotorsion pair, and it produces the characteristic module, which is both tilting and cotilting. The endomorphism algebra of the characteristic module is called the Ringel dual, which is again a quasi-hereditary algebra. Applying the Ringel dual twice, we obtain the original algebra again, up to Morita equivalence. I will explain these results following [Ringel 1991].

Y. Kimura : Quasi-hereditary algebras associated with reduced expressions in Coxeter groups

Let w be an element of the Coxeter group of a quiver. Buan-Iyama-Reiten-Scott studied a stably 2-Calabi-Yau category associated with w and constructed standard cluster tilting objects associated with reduced expressions of w. In this talk, we see that the endomorphism algebra of a standard cluster tilting object is a quasi-hereditary algebra, which is a result of Iyama-Reiten. Moreover, we have a presentation of the endomorphism algebra by a quiver with relations, which is a result of Buan-Iyama-Reiten-Smith.

R. Kodera : Representations of semisimple Lie algebras in the BGG category $\mathscr{O}$

Bernstein-Gelfand-Gelfand defined the category O in 1976, which is a full subcategory of the category of representations of a complex semisimple Lie algebra. It gives a typical example of highest weight category. I will start my first lecture with the definition of Lie algebras, and then explain the structure of semisimple Lie algebras (Cartan subalgebra and root space decomposition). In the second and third lectures, I review the highest weight theory for semisimple Lie algebras and properties of the category O.

H. Nagase : Recollements and Hochschild cohomology algebras

Hochschild cohomology is invariant under derived equivalences. On one hand, recollement of derived categories of algebras gives a long exact sequence of Hochschild cohomology of the algebras. In this talk, we will consider recollemnt of "module categories" of algebras and Hochschild cohomology of the algebras.

H. Minamoto : Quasi-hereditary algebras over rings and recollements

Highest weight categories and quasi-hereditary algebras over a field were introduced by Cline, Parshall and Scott.

Later they generalized the notion to more general commutative ground rings.

The purpose of this course is to explain them and in particular to explain Krause's recent result which characterize highest weight categories in terms of recollemant of abelian categories.

This formal treatment may clarify theory of highest weight category and quasi-hereditary algebras over a commutative ring.

We start the lecture by recalling the definition of homological recollement of abelian categories.

H.Miyachi : Level-rank duality

We summarise the level rank duality of quantum algebras in affine type A, and the categorification of this duality in terms of categories O's over rational double affine Hecke algebras, which are conjectured jointly by Joe Chuang and myself (ICRA and Nagoya'10). The major part of this conjecture was proved by Shan-Varagnolo-Vasserot and Rouquier-Shan-Varagnolo-Vasserot.

Since the contents are beyond my ability, perhaps I could only explain the situation.

K. Wada : Cellular algebras

A cellular algebra was introduced by Graham and Lehrer. It is an algebra with an anti-involution which gives a nice duality on its module category. Such duality naturally appears in Lie theory. In this lecture, I give definitions of cellular algebras and explain some fundamental properties of representations of cellular algebras. I will also explain connections with quasi-hereditary algebras.

M. Watanabe : Schubert polynomials, Kraskiewicz-Pragacz modules and quasi-hereditary algebras

In this course I will explain about an application of the theory of highest weight category to Schubert calculus, namely, an application on Kraskiewicz-Pragacz modules which arose from the study of Schubert polynomials.