Curled Algebra: Structure and Curvature

• Curled Algebra is a class of nonassociative algebras defined by the linear dependence between elements and their squares, often normalized as 0 or 1.
• It extends classical Koszul duality by incorporating explicit curvature data, which enables precise resolutions and computational advances in operadic and homotopy algebra.
• Curled algebras underpin structural classifications in low dimensions and drive insights in deformation theory via modified bar–cobar constructions and curved operator systems.

A curled algebra is a class of (typically nonassociative) algebras characterized by the strong linear dependence between generic elements and their squares. The term “curled” (sometimes used synonymously with “curved” or, in certain contexts, “curried”) designates algebraic structures where the square (or differential) fails to behave classically and instead is governed by a curvature functional, a scalar, or a revised compatibility. These concepts have been extensively formulated and analyzed in the context of Koszul duality, curved operads, Rota-Baxter systems, A∞-algebra deformations, nonassociative algebra classification, and representation theory of diagrammatic or combinatorial categories.

1. Foundational Definitions and Core Structure

A curled algebra ℱ over a field K is defined by the property that for every $x \in ℱ$, the elements $x$ and $x^2$ are linearly dependent: $\forall x \in ℱ, \,\, x^2 = ε_x x$ with $ε_x \in K$, and typically, after basis choices, the parameters $ε_x$ can be normalized to $0$ or $1$ (Takahasi et al., 28 Jul 2025). In the associative case, this property imposes significant restrictions, but curled algebras are generally nonassociative and include endo-commutative families, where the square map is a multiplicative endomorphism: $x^2 y^2 = (xy)^2$ for all $x, y \in ℱ$ (Takahasi et al., 2023, Takahasi et al., 28 Jul 2025).

Curved (a commonly interchangeable term, especially in homotopical and operadic contexts) algebras generalize differential graded (dg) algebras by replacing the square-zero condition on the differential with a controlled curvature: $$d^2(x) = μ(θ, x) - μ(x, θ) \quad \text{(associative case)}$$

$d^2(x) = [θ, x] \quad \text{(Lie case)}$

where θ is a curvature element (Lucio, 2022).

2. Curved Koszul Duality and the Role of Curvature

Curved Koszul duality extends classical duality theory to the setting where augmentation fails, necessitating explicit curvature data in dual structures (1008.5368, Idrissi, 2018, Zhang, 2021). For an operad or properad $\mathcal{P}$ with unit or constant terms (“quadratic-linear-constant” or QLC relations), the standard bar-cobar adjunction is replaced with a construction involving a “curved” cooperad $(C, d_C, θ)$, where the coderivation satisfies

$$d_C^2 = (θ \otimes \text{id}_C - \text{id}_C \otimes θ)\circ \Delta_{(1,1)}$$

and a twisting morphism α satisfies the modified Maurer–Cartan equation: $\partial(α) + α ★ α = Θ$ where the right side is zero in the classical theory (when θ = 0), but is nontrivial in the curved case (1008.5368, Zhang, 2021).

The curved Koszul dual coalgebra, arising from an algebra $A$ defined by QLC relations, is equipped with a coderivation and curvature map, and the cobar construction yields a small, explicit cofibrant resolution of $A$ even in the absence of augmentation. The curvature θ encodes the deviation from strict augmentation and allows the theory to resolve algebras with units and constant terms efficiently (Idrissi, 2018).

3. Bar–Cobar Constructions in the Curved Setting

Classically, the bar and cobar constructions establish an adjunction between (co)augmented (co)algebras. In the curved setting, the bar construction takes a unit-complemented curved algebra $A$ to a curved augmented coalgebra, and the cobar construction reverses this process (Lyubashenko, 2013). The key modifications include:

• The differential on the bar complex squares not to zero but to an inner derivation involving the curvature.
• Curved bar and cobar functors operate on categories of curved algebras and coalgebras with unit or counit splittings.
• The adjunction

$$\operatorname{Hom}_{\mathrm{UCCAlg}}(\operatorname{Cobar} C, A) \cong \operatorname{Hom}_{\mathrm{CACoalg}}(C, \operatorname{Bar} A)$$

generalizes classical bar–cobar duality to the curved context (Lyubashenko, 2013).

This extension is vital for explicit resolutions in the paper of homotopy theory, deformation theory, and the computation of derived invariants in nonaugmented settings.

4. Classification and Structural Analysis of Finite-Dimensional Curled Algebras

In low dimensions, curled algebras and their endo-commutative variants are subject to complete algebraic classification. For instance, in dimension two, a curled algebra is isomorphic to one of the following types (Takahasi et al., 2023):

• The trivial (zeropotent) algebra: $e^2 = 0, f^2 = 0, ef = 0, fe = 0$
• The anti-commutative algebra: $e^2 = 0, f^2 = 0, ef = e, fe = -e$
• One-parameter family: $e^2 = e, f^2 = 0, ef = a f, fe = (1-a)f$

Endo-commutativity in this context requires a set of algebraic conditions tying together structure coefficients to ensure the mapping $x \mapsto x^2$ is a multiplicative endomorphism.

In dimension three, necessary and sufficient conditions for endo-commutativity are derived in terms of the parameters specifying the multiplication table, codified as a system of quadratic and bilinear relations (labelled as equations (10) and (17) in (Takahasi et al., 28 Jul 2025)). These classify the exact circumstances under which a 3-dimensional curled algebra is endo-commutative.

5. Curvature and Deformation in Homotopy and Operadic Algebra

Curved (curled) structures underpin advances in homotopy algebraic frameworks. In the context of A∞-algebras, curvature emerges via two principal mechanisms (1101.2080):

• Deformation by a degree-1 element $y$ introduces $m_0^y \neq 0$, producing a curved A∞-algebra whose structure maps are sums over all insertions of $y$.
• Transfer of curvature along a chain contraction, producing a curved A∞-structure on a retracted complex, implicit in homotopy transfer and perturbation theory.

Curved operads and their calculus (including curved absolute operads and complete bar-cobar adjunctions) support universal constructions and a homotopical framework for curved algebraic objects (Lucio, 2022, Bellier-Millès et al., 2020). Notably, curved Koszul dual cooperads admit PBW-type isomorphisms to completions of classical cooperads for associated graded objects (Bellier-Millès et al., 2020), enabling explicit computations and linking curved and classical operadic invariants.

6. Curved Rota-Baxter and O-Operator Systems

Curved generalizations of Rota-Baxter systems incorporate an additional curvature term ω, modifying the defining identity to

$R(a)R(b) = R(R(a)b + aS(b)) + \omega(a \otimes b)$

with companion pre-Lie and associative structures induced under compatibility conditions on ω. In the case $R = S$ and $\omega$ bilinear, the Hochschild cochain complex over $A$ becomes a curved dg algebra, where $d^2(f) = [\omega, f]$ (Brzeziński, 2016).

Curved O-operator systems generalize curved Rota-Baxter systems and O-operators by considering curvature maps $w$ acting on bimodules, naturally interpolating between various algebraic systems (including dendriform, tridendriform, and pre-Lie algebras) through explicit construction and twisting (Ma et al., 2017).

7. Curried Algebras, Diagram Categories, and Representation Theory

In certain categorical and combinatorial settings, “curried” algebras (a usage of “curled” in the literature) reformulate classical Lie algebras to avoid explicit duals. For example, representations of $\mathfrak{gl}(V)$ can be recast as structure maps $a: V\otimes M \rightarrow V\otimes M$ satisfying specific compatibility relations (such as $[a_1, a_2] = \tau(a_1-a_2)$, with $\tau$ a symmetry involving the tensors) (Sam et al., 2022). This perspective elucidates equivalences between combinatorial categories (e.g., Brauer or partition categories) and categories of (curried) modules for classical Lie algebras, broadening the landscape of representation theory and absorbing diagrammatic combinatorics into algebraic structure.

8. Applications, Implications, and Theoretical Frameworks

Curled algebraic structures have broad applications:

• Resolution and homotopy transfer of unital associative and Frobenius algebraic structures, with direct relevance to 2d-TQFT and differential geometry (1008.5368).
• Development of explicit resolutions and derived functors (e.g., enveloping algebras, factorization homology) for Poisson n-algebras in topology and mathematical physics (Idrissi, 2018).
• Extension of cyclic (co)homology theories to the curved setting, generalizing the Feigin–Tsygan long exact sequence and supporting the paper of derived representation schemes (Zhang, 2021).
• Enrichment of deformation theory and homotopy algebra by encoding deformations and curvature-induced “twisting” via bar–cobar constructions and operadic calculus (Lucio, 2022, Lyubashenko, 2013).
• Classification and invariant theory for low-dimensional, nonassociative algebras, connecting to broader studies in nonassociative algebraic systems and variety degenerations (Takahasi et al., 2023, Takahasi et al., 28 Jul 2025).

9. Comparative Analysis and Unifying Perspective

While classical algebraic theories rely on augmentation or square-zero conditions, the curled (curved) framework systematically encodes failures of these as explicit structure—through curvature functionals, adjusted bar–cobar adjunctions, or explicit model category structures for curved operads. This paradigm enables:

• More compact and faithful resolutions for objects with units or constants.
• Homotopy-theoretic properties more naturally aligned with deformation phenomena.
• Broader unification of apparently disparate algebraic and combinatorial constructs under a twisted or curved algebraic umbrella.

This broadens classical methods, subsumes traditional Koszul duality, and provides computational and conceptual leverage in settings where curvature, augmentation failure, or diagrammatic complexities are intrinsic to the algebraic data.