Hamiltonian Systems on Quantum Spaces

TL;DR

This paper explores Hamiltonian systems on quantum spaces, revealing that Q-meromorphic Hamiltonians form a Virasoro algebra and Hamiltonian derivations form an $sl(2,A_1(q))$ Lie algebra, with motions linked to quadratic Hamiltonians.

Contribution

It demonstrates the algebraic structures of Hamiltonians and derivations on quantum spaces, connecting them to Virasoro and $sl(2,A_1(q))$ Lie algebras, and characterizes motions as quadratic Hamiltonians.

Findings

Q-meromorphic Hamiltonians form a Virasoro algebra with zero central charge.

Hamiltonian derivations form the Lie algebra $sl(2,A_1(q))$.

Any motion on a quantum space is associated with a quadratic Hamiltonian.

Abstract

In this paper we consider Hamiltonian systems on the quantum plane and we show that the set of Q-meromorphic Hamiltonians is a Virasoro algebra with central charge zero and the set of Hamiltonian derivations of the algebra of $Q$-analytic functions ${\cal A}_q$ with values in the algebra of $Q$-meromorphic functions ${\cal M}_q$ is the Lie algebra $sl(2,A_1(q)).$ Moreover we will show that any motion on a quantum space is associated with a quadratic Hamiltonian.