3-Dimensional Curled Algebra

• 3-Dimensional Curled Algebra is defined as a nonassociative system where every element x satisfies x² = λ(x)x, characterizing a degenerate squaring behavior.
• The structure is built on a three-element basis with type parameters (i, j, k) and mixed product constants that obey 21 explicit algebraic relations.
• Endo-commutativity, where (xy)² = x²y², imposes strict constraints on the structure, enabling classification and exploration of nearly idempotent algebraic behaviors.

A 3-Dimensional Curled Algebra is a nonassociative algebra of dimension three in which every element is linearly dependent with its square, i.e., for every $x$ in the algebra $\mathcal{A}$ there exists a scalar $\lambda(x)\in\mathbb{K}$ such that $x^2 = \lambda(x) x$. These structures arise in the paper of degenerate or idempotent-like behaviors in low-dimensional nonassociative algebras, and occupy a notable position in the landscape of finite-dimensional algebra classifications.

1. Definition and Algebraic Foundations

A curled algebra of dimension three over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $\mathcal{A}$ with a bilinear multiplication such that

$$x^2 \in \langle x \rangle, \quad \forall x \in \mathcal{A}.$$

There always exists a linear basis $\{e, f, g\}$ such that the basis squares are simple multiples of the basis vectors:

$e^2 = i\,e,\quad f^2 = j\,f,\quad g^2 = k\,g$

with $(i, j, k) \in \{0, 1\}^3$, and mixed products are expressed as $e f = A$, $e g = B$, $f e = C$, $f g = D$, $g e = E$, $g f = F$ with $A, B, C, D, E, F \in \mathcal{A}$. The structure constants of these mixed products encode the essential features of the algebra.

A key structural property is the relation between any $x$ and $x^2$. This property implies that all quadratic polynomials in $x$ reduce to linear polynomials, and that the multiplication table is highly degenerate in a sense reminiscent of idempotence but generalized to all nonzero elements.

2. Endo-Commutativity: Characterization and Main Theorem

A 3-dimensional curled algebra $\mathcal{A}$ is called endo-commutative if the (quadratic) squaring map is multiplicative, that is,

$$(x y)^2 = x^2 y^2, \qquad \forall x, y \in \mathcal{A}.$$

The endo-commutative property is strictly stronger than ordinary commutativity. The defining property imposes sharp constraints on the structure constants of the algebra.

The main result of (Takahasi et al., 28 Jul 2025) provides a necessary and sufficient condition for endo-commutativity in terms of the basis and the type $(i, j, k)$. Writing an element as $x = x_1 e + x_2 f + x_3 g$ and similarly $y$, one obtains:

• Condition (10): Eighteen equations involving the squares and symmetrized products of $A,B,C,D,E,F$ and the basis vectors, explicitly (listing a representative subset):
  • $A^2 = ij\,A$,
  • $B^2 = ik\,B$,
  • $C^2 = ij\,C$,
  • $D^2 = jk\,D$,
  • $E^2 = ik\,E$,
  • $F^2 = jk\,F$,
  • $A B + B A = i\,e(D + F)$,
  • $C E + E C = i(D + F)e$,
  • and similarly for the other nine identities.
• Condition (17): Three additional vectorial identities expressing the vanishing of certain off-diagonal terms:
  • $B C + B A + E C - A E = i(eF + F e)$,
  • $D A + F A + D C - C F = j(E f + f E)$,
  • $D B + F B + F E - E D = k(C g + g C)$.

A 3-dimensional curled algebra is endo-commutative if and only if all 21 equations above are satisfied for some basis $\{e, f, g\}$ of type $(i, j, k)\in\{0,1\}^3$ (Takahasi et al., 28 Jul 2025).

3. Structural Implications and Basis Dependence

These conditions are entirely determined by the structure constants relative to the chosen basis. The $(i, j, k)$ type specifies the behavior of the squares of basis vectors:

• $i = 1, j = 0, k = 1$ gives $e^2 = e$, $f^2 = 0$, $g^2 = g$;
• other combinations correspond to various mixtures of idempotent and zero-square elements.

The mixed products $A, B, C, D, E, F$ encapsulate all remaining freedom in multiplication and are subject to the 21 relations above. Thus, up to change of basis, an endo-commutative 3-dimensional curled algebra is completely determined by these parameters.

The type $(i, j, k)$ interacts crucially with the relations: if $i=0$, then all terms containing $ij$ or $ik$ vanish, simplifying some relations.

4. Relation to Other Nonassociative Structures

Curled algebras generalize zeropotent algebras (where $x^2 = 0$ for all $x$), but the endo-commutative subclass includes more general nonassociative examples. The necessary and sufficient condition unifies cases where the square mapping degenerates and gives new insight into the structure of degenerate or nearly-idempotent algebras.

Comparing to the results on skew-symmetric algebras (Remm, 2017) and 3-pre-Lie algebras (Ziying et al., 2023), 3-dimensional curled algebras highlight a different axis of structural degeneration, one which is not governed by commutator identities but by the algebraic behavior of the squaring process.

5. Examples and Applications

Given a basis $\{e, f, g\}$ with, for instance, $i = 1$, $j = 0$, $k = 1$, and setting $A = C = F = 0$, $B = E = D = g$, the necessary and sufficient conditions (10)-(17) reduce in such a way that the algebra becomes endo-commutative. Conversely, violating any of the equations, such as setting $B E \neq i (D + F) e$, yields an algebra that cannot be endo-commutative for any basis.

The explicit nature of the criteria allows classification and construction of all endo-commutative 3-dimensional curled algebras. Applications include:

• Detailed analysis of nilpotent and idempotent degeneracies in low dimensional algebraic geometry and combinatorial geometry,
• Testing endo-commutativity as an organizing property in the classification of nonassociative algebras,
• Exploration of possible deformations and extensions of algebraic structures arising in theoretical physics models that use degenerate or curled multiplication.

6. Broader Context and Significance

The necessary and sufficient characterization of endo-commutativity in three-dimensional curled algebras provides a concrete algebraic toolkit to address structure theory in nonassociative settings. Unlike the associative, Lie, or Jordan paradigms, the control over the squaring operation and its multiplicativity introduces new constraints and possibilities for degeneracy.

This suggests that further paper of these constraints (for example, in relation to automorphism groups or moduli spaces) could uncover new classes of nonassociative algebras and inform understanding of how algebraic degeneracies organize in finite dimensional settings.

The precise algebraic framework advanced in (Takahasi et al., 28 Jul 2025) enables systematic investigation, classification, and construction of 3-dimensional endo-commutative curled algebras, thereby contributing to the foundational understanding of nonassociative structures in mathematics.