[Paper Review] From forced gradings to Q-Koszul algebras

Brian Parshall, Leonard L. Scott|arXiv (Cornell University)|Feb 24, 2015

Algebraic structures and combinatorial models24 references3 citations

TL;DR

This paper introduces Q-Koszul algebras as a new class of graded algebras arising from forced gradings on quotients of the distribution algebra Dist(G) of a semisimple algebraic group G in positive characteristic. By constructing a positively graded algebra from ideal filtrations, the authors prove that the associated graded algebra is standard Q-Koszul, establishing a link between quantum group cohomology and algebraic group Ext groups via Kazhdan-Lusztig theory, especially for small primes and singular weights.

ABSTRACT

This paper has two parts. The main goal, carried out in Part I, is to survey some recent work by the authors in which "forced" grading constructions have played a significant role in the representation theory of semisimple algebraic groups $G$ in positive characteristic. The constructions begin with natural finite dimensional quotients of the distribution algebras Dist$(G)$, but then "force" gradings into the picture by passing to positively graded algebras constructed from ideal filtrations of these quotients. This process first guaranteed a place for itself by proving, for large primes, that all Weyl modules have $p$-Weyl filtrations. Later it led, under similar circumstances, to a new "good filtration" result for restricted Lie algebra Ext groups between restricted irreducible $G$-modules. In the process of proving these results, a new kind of graded algebra was invented, called a Q-Koszul algebra. Recent conjectures suggest these algebras arise in forced grading constructions as above, from quotients of Dist$(G)$, even for small primes and even in settings possibly involving singular weights. Related conjectures suggest a promising future for using Kazhdan-Lusztig theory to relate quantum and algebraic group cohomology and Ext groups in these same small prime and possibly singular weight settings. A part of one of these conjectures is proved in Part II of this paper. The proof is introduced by remarks of general interest on positively graded algebras and Morita equivalence, followed by a discussion of recent Koszulity results of Shan-Varagnalo-Vasserot, observing some extensions. Version 2 corrects some minor typos and inaccurate references. The paper will appear in PSPUM.

Motivation & Objective

• To develop a framework for studying the representation and cohomology theory of algebraic groups G in positive characteristic, particularly for small primes and singular weights.
• To address the lack of natural positive gradings in modular representation theory by introducing 'forced gradings' via ideal filtrations of finite-dimensional quotients of Dist(G).
• To define and investigate a new class of algebras—Q-Koszul and standard Q-Koszul algebras—that lie between quasi-hereditary and Koszul algebras.
• To establish a connection between quantum group Ext groups and algebraic group Ext groups using graded structures and Kazhdan-Lusztig polynomials.
• To prove that the Q-Koszul property is preserved under Morita equivalence and extend recent Koszulity results of Shan-Varagnolo-Vasserot to non-type A quantum groups.

Proposed method

• Construct a positively graded algebra gr̃A from a finite-dimensional quotient A of Dist(G) using ideal filtrations, thereby 'forcing' a grading where none naturally exists.
• Define Q-Koszul algebras as a generalization of Koszul algebras that interpolate between quasi-hereditary and Koszul structures, with a focus on Ext isomorphisms in the graded setting.
• Use Morita equivalence to show that the Q-Koszul property is invariant under equivalence of categories, enabling transfer of properties across derived categories.
• Leverage the parabolic-singular duality theory of Shan-Varagnolo-Vasserot to extend Koszulity results from type A to other quantum groups, under certain restrictions.
• Apply base change techniques in the graded setting to reduce conjectures about Ext groups in quantum and algebraic groups to isomorphisms involving the associated graded algebras.
• Utilize Kazhdan-Lusztig polynomials to conjecture explicit formulas for Ext dimensions in quantum group cohomology, with verification in the p-regular case for p > h.

Experimental results

Research questions

• RQ1 Can forced gradings on quotients of Dist(G) produce algebras with a Q-Koszul structure, even for small primes and singular weights?
• RQ2 To what extent does the Q-Koszul property of gr̃A imply isomorphisms between Ext groups in quantum and algebraic group settings?
• RQ3 How does the Q-Koszul property behave under Morita equivalence, and what does this imply for derived equivalences in representation theory?
• RQ4 Can the Koszulity of q-Schur algebras in non-type A settings be established using parabolic-singular duality theory?
• RQ5 Under what conditions does the algebra A itself admit a positive grading such that A ≅ gr̃A, and what are the implications for Ext group computations?

Key findings

• The Q-Koszul property is invariant under Morita equivalence, a result proven in Section 5 and foundational for transferring properties across equivalent categories.
• The associated graded algebra gr̃B of the quantum group algebra B = U_ζ,Γ is a standard Koszul algebra, as established in Theorem 7.2.
• For p-regular weights and p ≥ 2h−2 odd prime, the conjecture linking Ext groups in A and gr̃A holds, as shown via Theorem 3.4.
• When p > h and the Lusztig character formula holds, the conjecture on Ext isomorphisms is proved for p-regular weights, extending prior results from [CPS09].
• The Ext group dimensions in the quantum group setting are shown to match those in the graded algebra setting, with the equality preserved under base change to ℂ.
• Conjecture 7.3 reduces to a known result when Conjecture 4.6 holds, and the proof relies crucially on the Q-Koszul property enabling clean base change in the graded category.