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

Abstract: We study $2\times 2$ matrices over noncommutative rings with anti-involution, with a special focus on the symplectic group $\mathrm{Sp}_2(\mathcal{A},\sigma)$. 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 $\mathrm{Sp}_2(\mathcal{A},\sigma)$ with Manin matrices over general noncommutative rings; this naturally leads to a quantization $\mathrm{Sp}_2(\mathcal{A},\sigma)_q$. 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.