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$\mathrm{SL}_2$-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers

Zachary Greenberg, Dani Kaufman, Anna Wienhard

Abstract

We study $2\times 2$ matrices over noncommutative rings with anti-involution, with a special focus on the symplectic group $\mathrm{Sp}_2(\mathcal{A},σ)$. We define traces and determinants of such matrices and use them to prove a Cayley Hamilton identity and trace relations which generalize well known relations for elements of $\mathrm{SL}_2(R)$ over a commutative ring. We compare the structure of elements of $\mathrm{Sp}_2(\mathcal{A},σ)$ with Manin matrices over general noncommutative rings; this naturally leads to a quantization $\mathrm{Sp}_2(\mathcal{A},σ)_q$. In contrast to the usual definition of the quantum group as a deformation of the ring of matrix functions on $\mathrm{SL}_2(R)$, this quantization produces a group of matrices over a new noncommutative ring with involution. We finish the comparison by constructing a generalization of a Hopf algebra structure on the noncommutative ring of matrix functions of our quantum group. Finally, we use the noncommutative surface-type cluster algebras of Berenstein and Retakh to give a geometric interpretation of our Hopf algebra structure and to produce noncommutative generalizations of Markov numbers over many rings with involution including the complex numbers, dual numbers, matrix rings, and group rings.

$\mathrm{SL}_2$-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers

Abstract

We study matrices over noncommutative rings with anti-involution, with a special focus on the symplectic group . We define traces and determinants of such matrices and use them to prove a Cayley Hamilton identity and trace relations which generalize well known relations for elements of over a commutative ring. We compare the structure of elements of with Manin matrices over general noncommutative rings; this naturally leads to a quantization . In contrast to the usual definition of the quantum group as a deformation of the ring of matrix functions on , this quantization produces a group of matrices over a new noncommutative ring with involution. We finish the comparison by constructing a generalization of a Hopf algebra structure on the noncommutative ring of matrix functions of our quantum group. Finally, we use the noncommutative surface-type cluster algebras of Berenstein and Retakh to give a geometric interpretation of our Hopf algebra structure and to produce noncommutative generalizations of Markov numbers over many rings with involution including the complex numbers, dual numbers, matrix rings, and group rings.
Paper Structure (33 sections, 65 theorems, 149 equations, 10 figures)

This paper contains 33 sections, 65 theorems, 149 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Symplectic Groups over Noncommutative Rings
  4. Involutive Rings
  5. Basic Properties of the Symplectic Group
  6. Comparison with Manin Matrices
  7. The a sigma Plane
  8. Invariants of matrices
  9. Trace Identities
  10. Twisted Abelianization
  11. Quantization
  12. Quantum SL2
  13. Quantum Symplectic Group
  14. The Ring of Noncommutative Matrix Functions
  15. Involutive Algebras
  16. ...and 18 more sections

Key Result

Lemma 2.10

Let $\textsc{m} \in \mathop{\mathrm{Mat}}\nolimits_2(\mathcal{A},\sigma)$. We have the following Moreover, the adjoint of an element in $\mathrm{Sp}_2(\mathcal{A},\sigma)^l$ is an element of $\mathrm{Sp}_2(\mathcal{A},\sigma)^r$.

Figures (10)

  • Figure 1: Mutation at the arc $\gamma$.
  • Figure 2: Hexagon inside generic triangle.
  • Figure 3: Seed associated to a symplectic matrix.
  • Figure 4: Seeds associated to multiplying symplectic matrices.
  • Figure 5: Seed associated to transpose of a symplectic matrix.
  • ...and 5 more figures

Theorems & Definitions (219)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 209 more