##
**Duality in highest weight categories.**
*(English)*
Zbl 0701.18006

Classical groups and related topics, Proc. Conf., Beijing/China 1987, Contemp. Math. 82, 7-22 (1989).

[For the entire collection see Zbl 0658.00005.]

Let K be a field. A highest weight category is an abelian K-cateory \({\mathcal C}\) such that its Hom sets are finite dimensional vector K-spaces and each of its objects has a finite composition series [see the authors, J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)]. By a duality of \({\mathcal C}\) the authors mean an equivalence D: \({\mathcal C}\to {\mathcal C}^{op}\) such that \(D^ 2\cong Id_{{\mathcal C}}\) and D(S)\(\cong S\) for all simple objects S in \({\mathcal C}\). Highest weight categories with duality are investigated in the paper.

By definition every such category \({\mathcal C}\) has enough injective objects, there is a partially ordered set T and a bijection \(\lambda\mapsto S(\lambda)\) between T and the set of isoclasses of simple objects in \({\mathcal C}\) such that for each weight \(\lambda\in T\) the injective envelope I(\(\lambda\)) of S(\(\lambda\)) has a filtration \((*)\quad 0=F_ 0(\lambda)\subseteq F_ 1(\lambda)\subseteq...\subseteq F_{n(\lambda)}(\lambda)=I(\lambda)\) satisfying \(F_ 1(\lambda)\cong A(\lambda)\) and, for \(i>1\) \(F_ i(\lambda)/F_{i-1}(\lambda)\cong A(\mu_ i)\) for some \(\mu_ i\succ \lambda\). Here A(\(\lambda\)) is the largest subobject of I(\(\lambda\)) such that all composition factors S(\(\sigma\)) of I(\(\lambda\))/A(\(\lambda\)) satisfy \(\sigma\prec \lambda\). It is proved that if F: \({\mathcal C}\to {\mathcal C}'\) is a left exact functor such that \(FD\cong D'F\), the posets T and \(T'\) are equal and F maps A(\(\lambda\)) to \(A'(\lambda)\) for all \(\lambda\in T\) then F is an equivalence. Moreover, the following variant of the reciprocity result for \({\mathcal C}\) with duality is established. Let \(\lambda\),\(\mu\in T\) and let [I(\(\mu\)):S(\(\lambda\))] be the number of times A(\(\lambda\)) appears as a section \(F_ i(\mu)/F_{i-1}(\mu)\) in the filtration (*) of I(\(\mu\)) above. Then \([I(\lambda):A(\mu)]=[A(\mu):S(\lambda)].\)

It is also observed that if \({\mathcal C}\) is the category of finite dimensional modules over a quasi-hereditary algebra R then \({\mathcal C}\) is a highest weight category and any duality of \({\mathcal C}\) is induced by an involution of R which stabilizes any heredity chain of ideals in R. Several examples of highest weight categories with duality are given in the paper.

Let me add that highest weight categories with duality are also studied by R. S. Irving [“BGG algebras and the BGG reciprocity principle”, Inst. Adv. Stud., Princeton, Preprint (1988)].

Let K be a field. A highest weight category is an abelian K-cateory \({\mathcal C}\) such that its Hom sets are finite dimensional vector K-spaces and each of its objects has a finite composition series [see the authors, J. Reine Angew. Math. 391, 85-99 (1988; Zbl 0657.18005)]. By a duality of \({\mathcal C}\) the authors mean an equivalence D: \({\mathcal C}\to {\mathcal C}^{op}\) such that \(D^ 2\cong Id_{{\mathcal C}}\) and D(S)\(\cong S\) for all simple objects S in \({\mathcal C}\). Highest weight categories with duality are investigated in the paper.

By definition every such category \({\mathcal C}\) has enough injective objects, there is a partially ordered set T and a bijection \(\lambda\mapsto S(\lambda)\) between T and the set of isoclasses of simple objects in \({\mathcal C}\) such that for each weight \(\lambda\in T\) the injective envelope I(\(\lambda\)) of S(\(\lambda\)) has a filtration \((*)\quad 0=F_ 0(\lambda)\subseteq F_ 1(\lambda)\subseteq...\subseteq F_{n(\lambda)}(\lambda)=I(\lambda)\) satisfying \(F_ 1(\lambda)\cong A(\lambda)\) and, for \(i>1\) \(F_ i(\lambda)/F_{i-1}(\lambda)\cong A(\mu_ i)\) for some \(\mu_ i\succ \lambda\). Here A(\(\lambda\)) is the largest subobject of I(\(\lambda\)) such that all composition factors S(\(\sigma\)) of I(\(\lambda\))/A(\(\lambda\)) satisfy \(\sigma\prec \lambda\). It is proved that if F: \({\mathcal C}\to {\mathcal C}'\) is a left exact functor such that \(FD\cong D'F\), the posets T and \(T'\) are equal and F maps A(\(\lambda\)) to \(A'(\lambda)\) for all \(\lambda\in T\) then F is an equivalence. Moreover, the following variant of the reciprocity result for \({\mathcal C}\) with duality is established. Let \(\lambda\),\(\mu\in T\) and let [I(\(\mu\)):S(\(\lambda\))] be the number of times A(\(\lambda\)) appears as a section \(F_ i(\mu)/F_{i-1}(\mu)\) in the filtration (*) of I(\(\mu\)) above. Then \([I(\lambda):A(\mu)]=[A(\mu):S(\lambda)].\)

It is also observed that if \({\mathcal C}\) is the category of finite dimensional modules over a quasi-hereditary algebra R then \({\mathcal C}\) is a highest weight category and any duality of \({\mathcal C}\) is induced by an involution of R which stabilizes any heredity chain of ideals in R. Several examples of highest weight categories with duality are given in the paper.

Let me add that highest weight categories with duality are also studied by R. S. Irving [“BGG algebras and the BGG reciprocity principle”, Inst. Adv. Stud., Princeton, Preprint (1988)].

Reviewer: D.Simson

### MSC:

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |

16D90 | Module categories in associative algebras |

16P10 | Finite rings and finite-dimensional associative algebras |

18E10 | Abelian categories, Grothendieck categories |

20G15 | Linear algebraic groups over arbitrary fields |