[Paper Review] Noncommutative cross-ratio and Schwarz derivative

Vladimir Retakh, Vladimir Rubtsov+1 more|arXiv (Cornell University)|May 3, 2019

Advanced Operator Algebra Research15 references4 citations

TL;DR

This paper introduces a noncommutative generalization of the cross-ratio and Schwarz derivative using quasi-Plücker coordinates and quasideterminants over a noncommutative division ring. It establishes noncommutative analogues of classical geometric identities, including a noncommutative version of the pentagramma mirificum, and demonstrates that these structures underlie noncommutative integrable systems through recurrence relations with anti-periodic symmetry under an involution.

ABSTRACT

We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to "noncommutative elementary geometry" and relate it to noncommutative integrable systems. We also provide a noncommutative version of the celebrated "pentagramma mirificum".

Motivation & Objective

• To develop a noncommutative theory of cross-ratio and Schwarz derivative using quasi-Plücker coordinates and quasideterminants.
• To generalize classical geometric theorems such as Menelaus’s and Ceva’s theorems to noncommutative settings.
• To establish a noncommutative analogue of the pentagramma mirificum, including its recurrence relations and anti-periodic structure.
• To explore connections between noncommutative cross-ratios and noncommutative integrable systems, particularly higher pentagram maps and the Boussinesq hierarchy.
• To provide a foundation for future work on noncommutative integrable models and topological invariants.

Proposed method

• Define noncommutative cross-ratios via quasi-Plücker coordinates $ q_{ij}^k $ derived from quasideterminants of $ 2 \times n $ matrices over a noncommutative division ring $ \mathcal{R} $.
• Use quasideterminant identities to derive noncommutative versions of Plücker relations and skew-symmetry properties.
• Construct the noncommutative cross-ratio as $ \kappa(i,j,k,l) = q^j_{kl} q^i_{lk} $, with noncommutative inverses and transformation rules under involution.
• Derive recurrence relations for five vectors in $ \mathcal{R}^2 $, leading to a 5-antiperiodic system with $ x_{k+5} = \overline{x_k} $.
• Relate the noncommutative pentagramma to the continuous limit via the Boussinesq equation, suggesting a noncommutative integrable hierarchy.
• Apply the theory to noncommutative elementary geometry and link it to operator cross-ratios used in control theory.

Experimental results

Research questions

• RQ1 How can the classical cross-ratio and Schwarz derivative be generalized to noncommutative settings using quasi-Plücker coordinates?
• RQ2 What are the noncommutative analogues of Menelaus’s and Ceva’s theorems in terms of quasideterminants?
• RQ3 Can the classical pentagramma mirificum recurrence be extended to a noncommutative setting with anti-periodic symmetry?
• RQ4 What is the role of noncommutative cross-ratios in noncommutative integrable systems, particularly in higher pentagram maps?
• RQ5 Is there a noncommutative analogue of the Boussinesq equation arising from the continuous limit of noncommutative pentagram recurrences?

Key findings

• The noncommutative cross-ratio $ \kappa(i,j,k,l) = q^j_{kl} q^i_{lk} $ satisfies noncommutative skew-symmetry: $ q_{ij}^k q_{jk}^i q_{ki}^j = -1 $.
• The noncommutative Plücker identity holds: $ q_{ij}^k q_{ji}^\ell + q_{i\ell}^k q_{\ell i}^j = 1 $ for distinct indices.
• For five vectors in $ \mathcal{R}^2 $, the noncommutative pentagramma mirificum yields a 5-antiperiodic system with $ x_{k+5} = \overline{x_k} $.
• The recurrence relations are given by $ x_1 q^1_{32} x_3 q^1_{23} = 1 + x_2 $, and similar forms for other cyclic combinations.
• The system exhibits redundancy: relations with odd-indexed left-hand sides imply those with even-indexed ones, mirroring the commutative case.
• The continuous limit of such maps is conjectured to yield a noncommutative analogue of the Boussinesq equation, suggesting a link to noncommutative integrable hierarchies.