Noncommutative compact manifolds constructed from quivers

Lieven Le Bruyn^{∗} Universitaire Instelling Antwerpen B-2610 Antwerp (Belgium)

                lebruyn@wins.uia.ac.be

Abstract

The moduli spaces of θ-semistable representations of a finite quiver can be packaged together to form a noncommutative compact manifold.

If noncommutative affine schemes are geometric objects associated to affine associative ℂ-algebras, affine smooth noncommutative varieties ought to correspond to quasi-free (or formally smooth) algebras (having the lifting property for algebra morphisms modulo nilpotent ideals). Indeed, J. Cuntz and D. Quillen have shown that for an algebra to have a rich theory of differential forms allowing natural connections it must be quasi-free [1, Prop. 8.5]. M. Kontsevich and A. Rosenberg introduced noncommutative spaces generalizing the notion of stacks to the noncommutative case [5, §2]. It is hard to construct noncommutative compact manifolds in this framework, due to the scarcity of faithfully flat extensions for quasi-free algebras. An alternative was outlined by M. Kontsevich in [4] and made explicit in [5, §1] (see also [7] and [6]). Here, the geometric object corresponding to the quasi-free algebra A is the collection (rep_{n}A)_{n} where rep_{n}A is the affine GL_{n}-scheme of n-dimensional representations of A. As A is quasi-free each rep_{n}A is smooth and endowed with Kapronov’s formal noncommutative structure [2]. Moreover, this collection has equivariant sum-maps rep_{n}A × rep_{m}A rep_{m+n}A. We define a noncommutative compact manifold to be a collection (Y_{n})_{n} of projective varieties such that each Y_{n} is the quotient-scheme of a smooth GL_{n}-scheme X_{n} which is locally isomorphic to rep_{n}A_{α} for a fixed set of quasi-free algebras A_{α}; moreover, we require that the X_{n} are endowed with formal noncommutative structure, as well as with equivariant sum-maps X_{m} × X_{n} X_{m+n}. In this note we will construct of a large class of examples. An illustrative example : let M_{ℙ_{2}}(n; 0, n) be the moduli space of semi-stable vectorbundles of rank n over the projective plane ℙ_{2} with Chern-numbers c_{1} = 0 and c_{2} = n, then the collection (M_{ℙ_{2}}(n; 0, n))_{n} is a noncommutative compact manifold. In general, let Q be a quiver on k vertices without oriented cycles and let θ = (θ_{1}, …, θ_{k}) ∈ ℤ^{k}. For a finite dimensional representation N of Q with dimension vector α = (a_{1}, …, a_{k}) we denote θ(N) = ∑_{i} θ_{i}a_{i} and d(α) = ∑_{i} a_{i}. A representation M of Q is called θ-semistable if θ(M) = 0 and θ(N) ≥ 0 for every subrepresentation N of M. A. King studied the moduli spaces M_{Q}(α, θ) of θsemistable representations of Q of dimension vector α and proved that these are projective varieties [3, Prop 4.3]. We will prove the following result.

∗Research director of the FWO (Belgium)

Theorem 1 With notations as above, the collection of projective varieties

( ⨆_{d(α)=n} M_{Q}(α, θ) )_{n}

is a noncommutative compact manifold.

The claim about moduli spaces of vectorbundles on ℙ_{2} follows by considering the quiver • [figure] • and θ = (−1, 1). Let C be a smooth projective curve of genus g and M_{C}(n, 0) the moduli space of semi-stable vectorbundles of rank n and degree 0 over C. We expect the collection (M_{C}(n, 0))_{n} to be a noncommutative compact manifold.

1 The setting.

Let Q be a quiver on a finite set Q_{v} = {v_{1}, …, v_{k}} of vertices having a finite set Q_{a} of arrows. We assume that Q has no oriented cycles. The path algebra ℂQ has as underlying ℂ-vectorspace basis the set of all oriented paths in Q, including those of length zero which give idempotents corresponding to the vertices v_{i}. Multiplication in ℂQ is induced by (left) concatenation of paths. ℂQ is a finite dimensional quasi-free algebra. Let α = (a_{1}, …, a_{k}) be a dimension vector such that d(α) = n. Let rep_{Q}(α) be the affine space of α-dimensional representations of the quiver Q. That is,

rep_{Q}(α) = ⨁ ; _{•_{j}} a _{•_{i}} M_{a_{j}×a_{i}}(ℂ)

GL(α) = GL_{a_{1}} × … × GL_{a_{k}} acts on this space via basechange in the vertexspaces. For θ = (θ_{1}, …, θ_{k}) ∈ ℤ^{k} we denote with rep _{Q}^{ss}(α, θ) the open (possibly empty) subvariety of θ-semistable representations in rep_{Q}(α). Applying results of A. Schofield [8] there is an algorithm to determine the (α, θ) such that rep _{Q}^{ss}(α, θ) ≠ ∅. Consider the diagonal embedding of GL(α) in GL_{n} and the quotient morphism

X_{n} = ⨆_{d(α)=n} GL_{n} ×^{GL(α)} rep_{Q}^{ss}(α, θ) ^{π_{n}} [figure] Y_{n} = ⨆_{d(α)=n} M_{Q}(α, θ).

Clearly, X_{n} is a smooth GL_{n}-scheme and the direct sum of representations induces sum-maps X_{m} × X_{n} X_{m+n} which are equivariant with respect to GL_{m} × GL_{n} ^{⊂} GL_{m+n}. Y_{n} is a projective variety by [3, Prop. 4.3] and its points correspond to isoclasses of n-dimensional representations of ℂQ which are direct sums of θ-stable representations by [3, Prop. 3.2]. Recall that a θ-semistable representation M is called θ-stable provided the only subrepresentations N with θ(N) = 0 are M and 0.

2 Universal localizations.

We recall the notion of universal localization and refer to [9, Chp. 4] for full details. Let A be a ℂ-algebra and projmod A the category of finitely generated projective left A-modules. Let Σ be some class of maps in this category. In [9, Chp. 4] it is shown that there exists an algebra map A ^{j_{Σ}} A_{Σ} with the universal property that the maps A_{Σ} ⊗_{A} σ have an inverse for all σ ∈ Σ. A_{Σ} is called the universal localization of A with respect to the set of maps Σ. In the special case when A is the path algebra ℂQ of a quiver on k vertices, we can identify the isomorphism classes in projmod ℂQ with ℕ^{k}. To each vertex v_{i} corresponds an indecomposable projective left ℂQ-ideal P_{i} having as ℂ-vectorspace basis all paths in Q starting at v_{i}. We can also determine the space of homomorphisms

Hom_{ℂQ}(P_{i}, P_{j}) = ⨁_{•_{i} ........ p •_{j}} ℂp

where p is an oriented path in Q starting at v_{j} and ending at v_{i}. Therefore, any A-module morphism σ between two projective left modules

P_{i_{1}} ⊕ … ⊕ P_{i_{u}} ^{σ} P_{j_{1}} ⊕ … ⊕ P_{j_{v}}

can be represented by an u × v matrix M_{σ} whose (p, q)-entry m_{pq} is a linear combination of oriented paths in Q starting at v_{j_{q}} and ending at v_{i_{p}}. Now, form an v × u matrix N_{σ} of free variables y_{pq} and consider the algebra ℂQ_{σ} which is the quotient of the free product ℂQ ∗ ℂ⟨y_{11}, …, y_{uv}⟩ modulo the ideal of relations determined by the matrix equations

M_{σ}.N_{σ} = [\begin{matrix}v_{i_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & v_{i_{u}} \end{matrix} ] N_{σ}.M_{σ} = [\begin{matrix}v_{j_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & v_{j_{v}} \end{matrix} ] .

Repeating this procedure for every σ ∈ Σ we obtain the universal localization ℂQ_{Σ}. Observe that if Σ is a finite set of maps, then the universal localization ℂQ_{Σ} is an affine algebra. It is easy to see that ℂQ_{Σ} is quasi-free and that the representation space rep_{n} ℂQ_{σ} is an open subscheme (but possibly empty) of rep_{n} ℂQ. Indeed, if m = (m_{a})_{a} ∈ rep_{Q}(α), then m determines a point in rep_{n} ℂQ_{Σ} if and only if the matrices M_{σ}(m) in which the arrows are all replaced by the matrices m_{a} are invertible for all σ ∈ Σ. In particular, this induces numerical conditions on the dimension vectors α such that rep_{n}ℂQ_{Σ} ≠ ∅. Let α = (a_{1}, …, a_{k}) be a dimension vector such that ∑ a_{i} = n then every σ ∈ Σ say with

P_{1}^{⊕e_{1}} ⊕ … ⊕ P_{k}^{⊕e_{k}} ^{σ} P_{1}^{⊕f_{1}} ⊕ … ⊕ P_{k}^{⊕f_{k}}

gives the numerical condition e_{1}a_{1} + … + e_{k}a_{k} = f_{1}a_{1} + … + f_{k}a_{k}.

3 Local structure.

Fix θ = (θ_{1}, …, θ_{k}) ∈ ℤ^{k} and let Σ = ∪_{z∈ℕ_{+}}Σ_{z} where Σ_{z} is the set of all morphisms σ

P_{i_{1}}^{⊕zθ_{i_{1}}} ⊕ … ⊕ P_{i_{u}}^{⊕zθ_{iu}} ^{σ} P_{j_{1}}^{⊕−zθ_{j_{1}}} ⊕ … ⊕ P_{j_{v}}^{⊕−zθ_{jv}}

where {i_{1}, …, i_{u}} (resp. {j_{1}, …, j_{v}}) is the set of indices 1 ≤ i ≤ k such that θ_{i} > 0 (resp. θ_{i} < 0). Fix a dimension vector α with ⟨θ, α⟩ = 0, then θ determines a character χ_{θ} on GL(α) defined by χ_{θ}(g_{1}, …, g_{k}) = ∏ det(g_{i})^{θ_{i}}. With notations as before, the function d_{σ}(m) = det(M_{σ}(m)) for m ∈ rep_{Q}(α) is a semi-invariant of weight zχ_{θ} in ℂ[rep_{Q}(α)] if σ ∈ Σ_{z}. The open subset X_{σ}(α) = {m ∈ rep_{Q}(α) | d_{σ}(m) ≠ 0} consists of θ-semistable representations which are also n-dimensional representations of the universal localization ℂQ_{σ}. Under this correspondence θ-stable representations correspond to simple representations of ℂQ_{σ}. If we denote

X_{σ,n} = ⨆_{d(α)=n} GL_{n} ×^{GL(α)} X_{σ}(α) ^{⊂} X_{n}

then X_{σ,n} = rep_{n}ℂQ_{σ} and the restriction of π_{n} to X_{σ,n} is the GL_{n}-quotient map rep_{n}ℂQ_{σ} [figure] fac _{n}ℂQ_{σ} which sends an n-dimensional representation to the isomorphism class of the semi-simple n-dimensional representation of ℂQ_{σ} given by the sum of the Jordan-Hölder components, see [7, 2.3]. As the semiinvariants d_{σ} for σ ∈ Σ cover the moduli spaces M_{Q}(α, θ) this proves the local isomorphism condition for the collection (Y_{n})_{n}. A point y ∈ Y_{n} determines a unique closed orbit in X_{n} corresponding to a representation

M_{y} = M_{1}^{⊕e_{1}} ⊕ … ⊕ M_{l}^{⊕e_{l}}

with the M_{i} θ-stable representations occurring in M_{y} with multiplicity e_{i}. The local structure of Y_{n} near y is completely determined by a local quiver Γ_{y} on l vertices which usually has loops and oriented cycles and a dimension vector β_{y} = (e_{1}, …, e_{l}). The quiver-data (Γ_{y}, β_{y}) is determined by the canonical A_{∞}- structure on Ext_{ℂQ}^{∗}(M_{y}, M_{y}). As ℂQ is quasi-free, this ext-algebra has only components in degree zero (determining the vertices and the dimension vector β_{y}) and degree one (giving the arrows in Γ_{y}). Using [9, Thm 4.7] and the correspondence between θ-stable representations and simples of universal localizations, the local structure is the one outlined in [7, 2.5]. In particular, it can be used to locate the singularities of the projective varieties Y_{n}.

4 Formal structure.

In [2] M. Kapranov computes the formal neighborhood of commutative manifolds embedded in noncommutative manifolds. Equip a ℂ-algebra R with the commutator filtration having as part of degree −d

F_{−d} = ∑_{m} ∑_{i_{1}+...i_{m}=d} RR_{i_{1}}^{Lie}R … RR_{i_{m}}^{Lie}R

where R_{i}^{Lie} is the subspace spanned by all expressions [r_{1}, [r_{2}, […, [r_{i−1}, r_{i}] …] containing i − 1 instances of Lie brackets. We require that for R_{ab} = \frac{R}{F_{−1}} affine smooth, the algebras \frac{R}{F_{−d}} have the lifting property modulo nilpotent algebras in the category of d-nilpotent algebras (that is, those such that F_{−d} = 0). The micro-local structuresheaf with respect to the commutator filtration then defines a sheaf of noncommutative algebras on spec R_{ab}, the formal structure. Kapranov shows that in the affine case there exists an essentially unique such structure. For arbitrary manifolds there is an obstruction to the existence of a formal structure and when it exists it is no longer unique. We refer to [2, 4.6] for an operadic interpretation of these obstructions. We will write down the formal structure on the affine open subscheme rep_{n}ℂQ_{Γ} of X_{n} where Γ is a finite subset of Σ. Functoriality of this construction then implies that one can glue these structures together to define a formal structure on X_{n} finishing the proof of theorem 1. If A is an affine quasi-free algebra, the formal structure on rep_{n}A is given by the micro-structuresheaf for the commutator filtration on the affine algebra

\sqrt[n]{A} = A ∗ M_{n}(ℂ)^{M_{n}(ℂ)} = \{p ∈ A ∗ M_{n}(ℂ) | p.(1 ∗ m) = (1 ∗ m).p ∀m ∈ M_{n}(ℂ)\} This follows from the fact that \sqrt[n]{A} is again quasi-free by the coproduct theorems, [9, §2]. Specialize to the case when A = \sqrt[n]{ℂQ_{Γ}}. Consider the extended quiver Q̂(n) obtained from Q by adding one vertex v_{0} and by adding n arrows from v_{0} to v_{i}, for every vertex v_{i} in Q; these new arrows are denoted x_{i1}, …, x_{in}. Consider the morphism between projective left ℂQ̂(n)-modules

P_{1} ⊕ P_{2} ⊕ … ⊕ P_{k} ^{τ}  \underbrace{P_{0} ⊕ … ⊕ P_{0}}_{n}

determined by the matrix

M_{τ} = [\begin{matrix}x_{11} & … & … & x_{1n} \\ ⋮ & & & ⋮ \\ x_{k1} & … & … & x_{kn} \end{matrix}] .

Consider the universal localization B = ℂQ̂(n)_{Γ∪\{τ\}}. Then, \sqrt[n]{ℂQ_{Γ}} = v_{0}Bv_{0} the algebra of oriented loops based at v_{0}.

5 Odds and ends.

One can build a global combinatorial object from the universal localizations ℂQ_{Γ} with Γ a finite subset of Σ and gluings coming from unions of these sets. This example may be useful to modify the Kontsevich-Rosenberg proposal of noncommutative spaces to the quasi-free world. Finally, allowing oriented cycles in the quiver Q one can repeat the foregoing verbatim and obtain a projective space bundle over the collection (fac _{n}ℂQ)_{n}.

References

[1] J. Cuntz and D. Quillen. Algebra extensions and nonsingularity, Journal AMS 8 (1995) 251-289.

[2] M. Kapranov. Noncommutative geometry based on commutator expan-

sions, http://xxx.lanl.gov/math.AG/9802041 (1998) 48pp.

[3] A.D. King. Moduli of representations of finite dimensional algebras, Quat. J. Math. Oxford 45 (1994) 515-530.

[4] M. Kontsevich. Formal non-commutative symplectic geometry, in ’The Gelfand Mathematical Seminars, 1990-1992’, Birkhäuser (1993) 173-187.

[5] M. Kontsevich and A. Rosenberg. Noncommutative smooth spaces,

http://xxx.lanl.gov/math.AG/9812158 (1998) 18pp.

[6] M. Kontsevich. Non-commutative smooth spaces, talk at the Arbeitstagung-Bonn (1999)

[7] L. Le Bruyn. noncommutative geometry @_{n}, UIA-preprint 99-03

http://xxx.lanl.gov/math.AG/9904171 (1999) 14pp.

[8] A. Schofield. General representations of quivers. Proc. Lond. Math. Soc. 65 (1992) 46-64.

[9] A. Schofield. ’Representations of rings over skew fields’ London Math. Soc. Lecture Notes 92 (1995).

[AMA - Algebra Montpellier Announcements - 01-1999][November 1999]
Received July 21, 1999.