Reineke’s paper on Generic Extensions

The aim is to study the first four sections of the following paper, which is available on the internet at www.ams.org/ert .

M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147–163.

1 Introduction

Motivation.

1. Representations of Dynkin quivers. Gabriel’s Theorem says that the indecomposables are in bijection with the positive roots. In particular, representation finite. Auslander-Reiten theory tells us how to compute hom dimensions, and thus also ext dimensions. Representation directed. Problem: What are the extensions of representations?

2. Generic Ringel-Hall algebras. This is the ℤ[q]-algebra with basis the maps d : Δ_{+} → ℕ_{0} and structure constants the Hall polynomials. If we specialise to q a prime power, then this is isomorphic to the ℤ-algebra with basis the isomorphism classes of representations over the finite field 𝔽_{q} with structure constants given by counting numbers of extensions. These algebras are also related to the quantised enveloping algebras of the positive parts of semisimple Lie algebras. Again, we need to understand extensions.

This paper uses the geometry of representations to study extensions. In particular, given any two modules, there always exists a generic extension such that every extension of these two modules lies ‘inbetween’ the generic module and the direct sum. We shall show that taking the generic extension is an associate operation, so we can form a monoid M (with 1) and the monoid ring ℤM. We then relate this to the Ringel-Hall algebra.

Points to note. Form the generic Ringel-Hall algebra with parameter q.

1. When q is a prime power, we recover the above description involving the representations over the corresponding finite field.

2. When q = 1 we obtain the universal enveloping algebra of the positive part of the Lie algebra. This also has a description via the representations over an algebraically closed field of characteristic zero (using the Euler characteristic of flag varieties of modules).

3. When q = 0 we get the generic monoid ℤM. Again, this is defined over algebraically closed fields.

2 Representations of Dynkin quivers

• The Dynkin quivers of type 𝔸, 𝔻, 𝔼 and their representations (over an arbitrary field).

• Tit’s form ⟨−, −⟩, the symmetric form (−, −) and the positive roots Δ_{+}.

• Gabriel’s Theorem.

• Auslander-Reiten quiver.

• Representation directedness — can label the indecomposables X_{1}, …, X_{N} such that

Hom(X_{i}, X_{j}) = 0 for i > j and Ext(X_{i}, X_{j}) = 0 for i ≤ j.

M. Auslander, I. Reiten and S.O. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., vol. 36, Cambridge Univ. Press, 1995.

V.G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, 1990.

C.M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099, Springer-Verlag, 1984.

3 Geometry of representations

• Parametrising space Rep(Q, α, K) (with Zariski topology).

• Base change given by GL(α, K)-action.

• Orbits are irreducible and locally closed of dimension

dim O_{M} = dim GL(\underline{dim} M, K) − dim_{K} End(M).

The set \overline{O_{M}} −O_{M} is a union of orbits of strictly smaller dimension. Closed orbits correspond to semisimple representations. Open orbits correspond to representations without self-extensions (so includes all indecomposables).

• Representation finite implies every irreducible GL-stable set contains a dense orbit.

I have some notes by W. Crawley-Boevey that might be useful here.

4 Generic extensions

• Definition of E(M, N) for M, N GL-stable sets.

• Lemma 2.3 — which uses Steinberg’s Lemma: Let G be a connected affine algebraic group acting on a variety X. If Z ⊂ X is closed and stable under a parabolic P, then GZ is closed. Proof. Let Y = {(gP, x) | g^{−1}x ∈ Z}. This is (well-defined and) closed in G/P × X and its projection to X is GZ.

• Define M ⋆ N as being the dense orbit in E(O_{M}, O_{N}). This must have minimal dimension of endomorphism ring. So have geometric or algebraic description of generic extension.

Steinberg’s Lemma can be found in R. Steinberg, Conjugacy Classes in Algebraic Groups, Lecture Notes in

Math., vol. 366, Springer-Verlag, 1980.

5 Degenerations

• Definitions of the partial orders ≤_{deg} and ≤_{hom} (latter is a symmetric definition). Clearly have cancellation in ≤_{hom}.

• (Zwara) Representation finite implies these partial orders coincide.

• (Bongartz) If M, N are indecomposable with no path M → N in the ARquiver, then X ≤_{deg} M ⊕ N iff there is an extension 0 → N → X → M → 0, and there exists such an X without self-extensions (so generic).

• Deduce that the middle term of an AR-sequence is always the generic extension. Furthermore, can read off generic extension from the AR-quiver.

• Examples — 𝔸_{4}, 𝔻_{4}. K. Bongartz, On degenerations and extensions of finite dimensional modules,

Adv. Math. 121 (1996), 245–287.

G. Zwara, Degenerations for modules over representation-finite algebras, Proc. Amer. Math. Soc. 127 (1999), 1313–1322.

6 Associativity of ⋆

• Know that if 0 → N → X → M → 0 then M ⋆ N ≤_{deg} X ≤_{deg} M ⊕ N.

• Converse is true if M, N are indecomposable (previous talk), but false in general (example b in Section 2).

• Proposition 2.4 — which says \overline{E(O_{M}, O_{N})} = E(\overline{O_{M}}, \overline{O_{N}}).

• Lemma 3.1.

• Definition 3.2.

7 Generators and relations

• Proprosition 3.3 — M is generated by the simples.

• Lemma 3.4 — the fundamental relations.

• Algorithm for computing arbitrary generic extensions.

• Example???

8 Ringel-Hall algebras

• Definition of H_{q}(Q) for representations over the finite field 𝔽_{q}.

• Riedtmann’s formula for structure constants

g_{MN}^{X} = q^{−[M,N]} \frac{a_{X}}{a_{M}a_{N}} |\{0 → N → Y → M → 0 | Y ≅ X\}|

• (Ringel) The g_{MN}^{X} are given by evaluation at q of an integer polynomial. So can define generic Ringel-Hall algebra H(Q).

• Presentation by generators and relations. Generators are the simples [S_{i}] with relations

[S_{i}] ⋄ [S_{j}] − [S_{j}] ⋄ [S_{i}] = 0 if no edge between i and j

and for i → j we have

[S_{i}] ⋄ [S_{i}] ⋄ [S_{j}] − (q + 1)[S_{i}] ⋄ [S_{j}] ⋄ [S_{i}] + q[S_{j}] ⋄ [S_{i}] ⋄ [S_{i}] = 0
[S_{i}] ⋄ [S_{j}] ⋄ [S_{j}] − (q + 1)[S_{j}] ⋄ [S_{i}] ⋄ [S_{j}] + q[S_{j}] ⋄ [S_{j}] ⋄ [S_{i}] = 0.

C. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994), 526–546.

C.M. Ringel, The Hall Algebra Approach to Quantum Groups, E.L.A.M. Lectures 1993, Aportaciones Matemáticas Comunicaciones 15 (1995), 85–114. The latter is available on the internet at www.mathematik.uni-bielefeld.de/birep and follow the link to summer schools.

9 Isomorphism

• Theorem 4.2 — using specialisation is compatible with presentation by generators and relations.

• Note that although simples correspond, other representations do not. However, get a basis of H_{0}(Q) which is closed under multiplication.

• Also see that H_{1}(Q) is isomorphic to U(𝔤(Q)_{+}).

• Mention Schofield’s approach to constructing U(𝔤(Q)_{+}) via Euler characteristic.

Where can we go from here?

• Two parameter quantum groups and their canonical bases.

• Extension to all quivers using ‘families of quiver representations’ and general representations.

• Extension to non-symmetric Dynkin diagrams using quivers with automorphism.