[Paper Review] Naturally graded Zinbiel algebras with nilindex $n-3$

J. Q. Adashev, L.M. Camacho+2 more|arXiv (Cornell University)|Mar 23, 2013

Algebraic structures and combinatorial models3 references4 citations

TL;DR

This paper completes the classification of complex $n$-dimensional naturally graded Zinbiel algebras with nilindex $n-3$ by fully characterizing those with characteristic sequence $(n-3,2,1)$. Using structural analysis and the Zinbiel identity, the authors prove the existence of a unique algebra of this type for $n \geq 8$, while showing non-existence for type II and III algebras when $n \geq 9$ and $n \geq 7$, respectively, via contradiction from Zinbiel identity constraints.

ABSTRACT

We present the classification of a subclass of $n$-dimensional naturally graded Zinbiel algebras. This subclass has the nilindex $n-3$ and the characteristic sequence $(n-3,2,1).$ In fact, this result completes the classification of naturally graded Zinbiel algebras of nilindex $n-3.$

Motivation & Objective

• To complete the classification of complex $n$-dimensional naturally graded Zinbiel algebras with nilindex $n-3$ by resolving the remaining case of characteristic sequence $(n-3,2,1)$.
• To analyze the structural constraints imposed by the Zinbiel identity on algebras with this nilindex and characteristic sequence.
• To prove the non-existence of such algebras of type II for $n \geq 9$ and type III for $n \geq 7$ using consistency conditions derived from the Zinbiel identity.
• To provide a complete structural characterization of the unique algebra of type I with characteristic sequence $(n-3,2,1)$ for $n \geq 8$.
• To extend prior classifications of Zinbiel algebras with higher nilindex, completing the program for nilindex $n-3$.

Proposed method

• The authors use the Zinbiel identity $ (x \circ y) \circ z = x \circ (y \circ z) + x \circ (z \circ y) $ as the fundamental algebraic constraint to derive relations among basis elements.
• They classify algebras by their characteristic sequence $ C(\mathcal{Z}) = \max\{C(x) \mid x \in \mathcal{Z} \setminus \mathcal{Z}^2\} $, focusing on $ (n-3,2,1) $, and define the natural grading via the descending sequence of Jordan block dimensions of left multiplication operators.
• For each algebra type (I, II, III), they construct a basis and derive products using the Zinbiel identity, leading to systems of equations in structure constants.
• They employ symbolic computation via Mathematica to verify and generalize low-dimensional calculations and to test consistency of structure constants.
• For type II and III algebras, they derive contradictions from overdetermined systems of equations arising from the Zinbiel identity, proving non-existence for $n \geq 9$ and $n \geq 7$, respectively.
• The proof for type I relies on solving the Zinbiel identity constraints, showing that all structure constants must vanish except for specific non-zero products, leading to a unique algebra structure.

Experimental results

Research questions

• RQ1 Does a naturally graded Zinbiel algebra of nilindex $n-3$ with characteristic sequence $(n-3,2,1)$ exist for $n \geq 8$?
• RQ2 What are the structural constraints imposed by the Zinbiel identity on such algebras, and how do they determine the possible non-zero products?
• RQ3 Why do type II naturally graded Zinbiel algebras with $n \geq 9$ and type III with $n \geq 7$ fail to exist under the given nilindex and characteristic sequence?
• RQ4 Can the classification of $n$-dimensional naturally graded Zinbiel algebras with nilindex $n-3$ be completed by resolving the $(n-3,2,1)$ case?
• RQ5 What is the unique algebra structure for the $(n-3,2,1)$ case, and how is it characterized by its products and grading?

Key findings

• A unique naturally graded Zinbiel algebra of type I with characteristic sequence $(n-3,2,1)$ exists for all $n \geq 8$, and its structure is fully determined by the non-zero products $e_1 \circ e_1 = e_2$, $e_1 \circ e_i = e_{i+1}$ for $3 \leq i \leq n-2$, $e_1 \circ e_{n-1} = e_n$, and $e_1 \circ e_n = 0$, with all other products zero.
• For type II algebras, the system of equations derived from the Zinbiel identity $Z(e_1,e_2,e_3)=Z(e_1,e_2,e_4)=Z(e_1,e_2,e_5)=0$ leads to an inconsistent system: $1+\beta_1+\beta_2=0$, $3+2\beta_1+\beta_2=0$, $6+3\beta_1+\beta_2=0$, which has no solution, proving non-existence for $n \geq 9$.
• For type III algebras, the Zinbiel identity forces $e_1 \circ e_4 = e_5$, but $e_1 \circ e_4 = e_5$ contradicts the requirement that $e_5 \in Z_4$ and $e_1 \circ e_4 = e_5$ implies $e_5 = 0$ when $n \geq 6$, leading to a contradiction and proving non-existence for $n \geq 7$.
• The classification of naturally graded Zinbiel algebras with nilindex $n-3$ is now complete, as all three possible characteristic sequences $(n-3,3)$, $(n-3,1,1,1)$, and $(n-3,2,1)$ have been resolved.
• The unique algebra of type I with characteristic sequence $(n-3,2,1)$ has $\dim \mathcal{Z}^{n-3} = 1$, $\dim \mathcal{Z}^{n-2} = 1$, and $\dim \mathcal{Z}^{n-1} = 1$, consistent with nilindex $n-3$.
• All structure constants in the type I algebra are zero except for the specified non-zero products, and the Zinbiel identity is satisfied for all basis elements, confirming the algebra's validity.