Crawley-Boevey's Conjectural Solution

• Crawley-Boevey's Conjectural Solution is a framework establishing minimal noncommutative structures that yield Poisson brackets on representation spaces via canonical trace maps.
• It extends H₀-Poisson structures to derived settings using cyclic homology and semifree DG algebra resolutions.
• The approach has practical applications in moduli theory, notably addressing the Deligne–Simpson problem through quiver varieties and homotopy invariance.

Crawley-Boevey’s conjectural solution is a set of influential ideas and results concerning the minimal structure required on noncommutative algebras to induce Poisson structures on representation spaces, with deep consequences for noncommutative geometry, representation theory, and the theory of moduli spaces. The conjectural solution has been confirmed and extended in several directions, most notably by the construction of derived Poisson structures that generalize the original setup to higher homological levels, and by applications in the solution of the Deligne–Simpson problem for moduli of local systems via quiver varieties. The central theme is the characterization, transfer, and extension of Poisson structures from noncommutative settings to commutative representation-theoretic moduli, leading to new homotopy-invariant frameworks and compatibility results across classical and derived contexts.

1. Origins: The Kontsevich Principle and H₀-Poisson Structures

The foundational principle arises from Kontsevich’s viewpoint on inducing commutative Poisson geometry on representation spaces of associative algebras $A$ by equipping $A$ with a minimal "noncommutative" Poisson structure. Crawley-Boevey defined an $H_{0}$-Poisson structure as a Lie bracket on the abelianization $A/[A,A]$ satisfying compatibility conditions with trace maps

$$\operatorname{Tr}_{V}: A/[A,A] \rightarrow k[\operatorname{Rep}_V(A)]^{GL(V)}$$

such that the induced bracket descends to a genuine Poisson bracket on the function algebra of the representation scheme. This construction is designed to be the weakest possible, capturing only the essential commutator-level information needed for derived geometric transfer (Berest et al., 2012).

2. Extension to Derived Poisson Structures

The principal advancement is the generalization of the $H_{0}$-Poisson structure to a graded (super) Lie algebra structure on the full cyclic homology

$$\{-, -\}: \mathrm{HC}_{\bullet}(A) \times \mathrm{HC}_{\bullet}(A) \longrightarrow \mathrm{HC}_{\bullet}(A)$$

This derived Poisson structure, termed "derived Poisson structure" (Editor's term), arises naturally on cofibrant resolutions (semifree DG algebras) of $A$. The restriction to degree zero recovers Crawley-Boevey’s original bracket. The key homotopical property is that homotopy equivalent noncommutative Poisson structures on $A$ induce, via the derived representation functor, homotopy equivalent Poisson algebra structures on the derived representation scheme

$$\mathrm{Tr}_V(\{a, b\}) = \{\mathrm{Tr}_V(a), \mathrm{Tr}_V(b)\},\quad a, b \in \mathrm{HC}_{\bullet}(A)$$

This ensures quasi-isomorphism invariance, incorporating higher homological data absent in degree zero (Berest et al., 2012).

3. Explicit Constructions via Cobar and Double Poisson Brackets

Derived Poisson structures are concretely realized via the cobar construction $\Omega(C)$, where $C$ is a cyclic or Calabi–Yau DG coalgebra (often the linear dual of a finite-dimensional cyclic algebra). Van den Bergh’s double Poisson bracket yields a $(2-n)$-double Poisson bracket on $\Omega(C)$, inducing a derived noncommutative Poisson structure on $A$ when $\Omega(C)$ resolves $A$. Explicit formulas, such as

$$\{v, w\} = \sum_{i, j} \pm (w_1,\dots, w_{j-1}, v_{i+1},\dots,v_n) \otimes (v_1,\dots, v_{i-1}, w_{j+1},\dots, w_m)$$

for $v = (v_1,\dots,v_n), w = (w_1,\dots, w_m)$, with signs from the Koszul rule, provide calculable models, linking cyclic, Calabi–Yau, and string topology frameworks (Berest et al., 2012).

4. Applications: Deligne–Simpson Problem and Representation Moduli

The conjectural solution finds geometric application in the Deligne–Simpson problem, where necessary conditions for the existence of irreducible local systems (solutions to $A_1\cdots A_k = 1$ in prescribed conjugacy classes) are reduced to combinatorial root-theoretic constraints on dimension vectors of star-shaped quivers: $$d \in \Sigma_q = \left\{ d \in R_q^+\, \middle|\, \text{if } d = \sum_s d^{(s)},\, p(d) > \sum_s p(d^{(s)}) \right\}$$ with $R_q^+$ the set of positive roots of a Kac–Moody algebra for given parameters $q$, and $p(d)$ the defect function. The necessity result employs the nonabelian Hodge correspondence, variation of parabolic weights, and root-theoretic reductions (using results of Schedler–Tirelli and Kostov) to establish that irreducible solutions exist only when $d$ satisfies these conditions, confirming the necessity half of Crawley-Boevey’s conjectural solution for the tame case (Shu, 15 Sep 2025).

5. Homotopy Invariance and Trace Maps

A fundamental feature distinguishing the derived approach is the homotopy invariance encoded in the passage from noncommutative to commutative Poisson structures via trace maps. The derived structure

$$\operatorname{Tr}_{V}\left( \{ a, b \} \right) = \{ \operatorname{Tr}_V(a), \operatorname{Tr}_V(b) \}$$

extends classical compatibility to higher homological degrees, integrating the trace maps into a homotopically coherent system that respects quasi-isomorphisms and serves as a bridge between noncommutative and commutative geometry. This renders computations in derived representation schemes robust under homological deformations and supports compatibility of Poisson structures across levels (Berest et al., 2012).

6. Structural and Computational Implications

The approach provides explicit methods for constructing and identifying derived Poisson structures using semifree DG resolutions and Van den Bergh’s double bracket formalism. The framework is highly amenable to computational techniques through explicit formulas on cobar DG algebras and their homology, effectively connecting the combinatorics of quiver representations, the geometry of moduli spaces, and the algebra of Poisson brackets.

Algebraic Structure	Poisson Structure Induced	Extension Mechanism
$A/[A,A]$ (abelianization)	$H_0$-Poisson (Lie bracket)	Trace maps to $k[\operatorname{Rep}_V(A)]$
$\Omega(C)$ (cobar DG algebra)	Derived $(2-n)$-Poisson structure	Double bracket, cyclic homology
$\mathrm{HC}_\bullet(A)$	Graded Poisson via derived bracket	Cofibrant resolution, trace maps

7. Broader Impact and Connections

The theory situates Crawley-Boevey’s conjectural solution within the context of noncommutative geometry, representation theory, and moduli of spaces of local systems. Its homotopy-theoretic perspective connects to string topology, Calabi–Yau algebras, and the homology of representation varieties. The methods and results can be applied to paper symplectic and Poisson moduli in derived and noncommutative settings, offering computational tools for explicit calculations in derived categories and representation schemes. The synthesis of algebraic, geometric, and homological techniques defines a conceptual and technical framework for advancing the paper of moduli spaces and their invariants.