[Paper Review] $q$-Deformations and $t$-deformations of Markov triples

Takeyoshi Kogiso|arXiv (Cornell University)|Aug 29, 2020

Advanced Topics in Algebra6 references4 citations

TL;DR

This paper introduces two distinct generalizations of Markov triples: $q$-deformations using $q$-continued fractions from cluster algebras and quantum topology, and $t$-deformations via castling transformations of prehomogeneous vector spaces. It establishes a precise correspondence between them through the identity $f_w([3]_q/q) = q h_w(q)$, unifying quantum algebra and representation theory through Markov-type equations.

ABSTRACT

In this paper, we generalize the Markov triples in two different directions. One is generalization in direction of using the $q$-deformation of rational number introduced by \cite{MO} in connection with cluster algebras, quantum topology and analytic number theory. The other is direction using castling transforms of prehomogeneous vector spaces \cite{SaKi} which plays an important role in the study of representation theory and automorphic function. In addition, the present paper gives a relationship between the two generalizations. This may provide some kind of bridging between different fields.

Motivation & Objective

• To generalize Markov triples using $q$-deformations of rational numbers via continued fractions, extending classical Markov theory into quantum and cluster algebra frameworks.
• To introduce $t$-deformations of Markov triples through castling transformations of prehomogeneous vector spaces, linking representation theory and automorphic forms.
• To establish a deep algebraic correspondence between $q$-deformations and $t$-deformations, revealing a unifying structure across disparate mathematical fields.
• To explore whether $q$-deformations yield analogous approximation properties for $q$-quadratic irrationals, extending classical Diophantine approximation.

Proposed method

• Define $q$-deformed rational numbers $[r/s]_q$ using two-sided continued fraction expansions with alternating $q$-integers and $q$-powers, ensuring consistency between regular and negative continued fractions.
• Construct $q$-deformed Markov triples as traces of $q$-deformed word evaluations in $SL(2,\mathbb{Z}[q,q^{-1}])$ matrices $A_q$ and $B_q$, yielding solutions to a $q$-deformed Markov equation with correction term $\frac{(q-1)^2}{q^3}$.
• Apply castling transformations to $t$-dimensional prehomogeneous vector spaces to generate $t$-deformed Markov triples as polynomial invariants $f_w(t)$, satisfying a $t$-deformed Markov equation with constant $t-3$.
• Derive the key identity $f_w([3]_q/q) = q h_w(q)$ by substituting $t = q^{-1}[3]_q$ into the $t$-deformed equation and matching it to the $q$-deformed equation, proving equivalence of the two frameworks.
• Use explicit matrix constructions and trace identities to verify that $q$-deformed Markov triples satisfy $x^2 + y^2 + z^2 + \frac{(q-1)^2}{q^3} = [3]_q xyz$ and $t$-deformed ones satisfy $x^2 + y^2 + z^2 + (t-3) = txyz$.
• Verify the correspondence via direct computation on examples such as $f_{a^2b}(t)$ and $h_{a^2b}(q)$, showing $f_{a^2b}(q^{-1}[3]_q) = q h_{a^2b}(q)$.

Experimental results

Research questions

• RQ1 Can $q$-deformations of Markov triples provide a meaningful approximation theory for $q$-quadratic irrationals, analogous to classical Markov theory?
• RQ2 Is there a deeper structural link between quantum topology, analytic number theory, and prehomogeneous vector spaces through the $q$-$t$ correspondence?
• RQ3 How do castling transformations of prehomogeneous vector spaces generate polynomial invariants that mirror the structure of $q$-deformed Markov triples?
• RQ4 Does the identity $f_w([3]_q/q) = q h_w(q)$ reflect a universal phenomenon across all Christoffel words, or is it limited to specific cases?

Key findings

• The $q$-deformed Markov triple $(h_w(q), h_{ww'}(q), h_{w'}(q))$ satisfies the equation $x^2 + y^2 + z^2 + \frac{(q-1)^2}{q^3} = [3]_q xyz$, generalizing the classical Markov equation.
• The $t$-deformed Markov triple $(f_w(t), f_{ww'}(t), f_{w'}(t))$ satisfies the equation $x^2 + y^2 + z^2 + (t-3) = t xyz$, with $f_w(t)$ being monic integer polynomials of increasing degree.
• The correspondence $f_w([3]_q/q) = q h_w(q)$ holds for all Christoffel words $w$, establishing a precise algebraic bridge between $q$-deformations and $t$-deformations.
• Explicit computations confirm the identity for $f_{a^2b}(t)$ and $f_{a^2bab}(t)$, showing $f_{a^2b}(q^{-1}[3]_q) = q h_{a^2b}(q)$ and $f_{a^2bab}(q^{-1}[3]_q) = q h_{a^2bab}(q)$.
• The $q$-deformed matrices $A_q$ and $B_q$ are in $SL(2,\mathbb{Z}[q,q^{-1}])$, ensuring integrality and compatibility with trace-based constructions.
• The $t$-deformed polynomials $f_w(t)$ are generated via castling transformations on prehomogeneous vector spaces, and their tree structure mirrors the Markov tree of classical triples.