Characterizing algebraic groups via maximal tori

理解对称性，以及它如何以及为什么在自然界中出现，在数学和物理学中都很重要。回想一下，测量对称性的数学对象被称为“群”，他们的研究被称为“群论”。群的经典例子是旋转或平移（连续群）和正方形或雪花对称（离散群）。世纪中期，代数群诞生了，代数群是一种具有连续群（所谓的李群）和离散群的精神的对象，但它更具有普遍性。在随后的几十年中，代数群理论被用来统一处理代数和数论的几个关键领域，包括二次型理论、中心单代数、对合代数和一些非结合代数。

Understanding symmetry, and how and why it arises in nature, is important in both Mathematics and Physics. Recall that the mathematical objects that measure symmetry are called ``groups'' and their study is known as ``group theory''. Classical examples of groups are those of rotations or translations (continuous groups) and the symmetry of a square or a snowflake (discrete groups). The mid 20th century saw the birth of Algebraic Groups, objects that capture the spirit of continuous groups (the so-called Lie groups) and discrete groups, but that are much more universal. Over the course of subsequent decades the theory of algebraic groups has been used to give a unified treatment of several key areas of algebra and number theory, including the theories of quadratic forms, central simple algebras, algebras with involution and some non-associative algebras.