# The conformal dilatation and Beltrami forms over quadratic field extensions

## TL;DR

This paper develops an algebraic framework for complex dilatation and Beltrami forms over quadratic field extensions, generalizing classical geometric notions to a broader algebraic context with invariant properties.

## Contribution

It introduces an algebraic analogue of complex dilatation and Beltrami forms over quadratic field extensions, providing an invariant and generalized approach.

## Key findings

- Establishes a relationship between conformal dilatation and Beltrami form over quadratic extensions.
- Shows that working with general field extensions clarifies algebraic aspects without added difficulty.
- Provides an invariant version of the classical geometric approach to complex dilatation.

## Abstract

The paper is devoted to an algebraic analogue of a geometric approach to the classical notion of complex dilatation suggested in the paper arXiv:1701.06259 [math.CV] by the author. At the same time it provides an invariant version of this geometric approach. From the algebraic point of view it is only natural to work with a general field extension K/k of degree 2 instead of the fields of real and complex number (under the assumption that the characteristic is not equal to 2). Given a k-linear map between two K-vector spaces of dimension 1 over K, there are two natural measures of deviation of this map from being K-linear: its conformal dilatation, defined in terms of quadratic forms over k, and its Beltrami form, directly generalizing the classical complex dilatation. It turns out that these two measures are related in the same way as in the classical case. Working with a general field extension does not lead to any new difficulties compared to the classical case, but only clarifies the algebraic aspects of the theory.