# A class of differential quadratic algebras and their symmetries

## TL;DR

This paper investigates a family of quadratic algebras with four generators, exploring their symmetries and differential structures, including notable special cases like Sklyanin algebras and noncommutative spheres.

## Contribution

It introduces a broad class of quadratic algebras, determines their quantum symmetry groups, and constructs covariant differential calculi, unifying several known noncommutative geometries.

## Key findings

- Identified quantum groups of symmetries for the algebras
- Constructed finite-dimensional covariant differential calculi
- Included special cases like Sklyanin algebras and noncommutative spheres

## Abstract

We study a multi-parametric family of quadratic algebras in four generators, which includes coordinate algebras of noncommutative four-planes and, as quotient algebras, noncommutative three spheres. Particular subfamilies comprise Sklyanin algebras and Connes--Dubois-Violette planes. We determine quantum groups of symmetries for the general algebras and construct finite-dimensional covariant differential calculi.