Study of algebras relating to quadratic form by using representation theory and homological algebras

2004年7月24日至26日，我们参加了在中国南京举行的第四届日-中-韩代数理论国际研讨会和第十一届代数表示理论国际会议，介绍了最近的研究成果，并观察了研究趋势。2005年，我们在爱知工业大学举办了第38届环与表示理论研讨会，我们邀请了与我们研究密切相关的Raphael Rouquier教授（University Denis Diderot-Paris 7），请他讲了解决Broue猜想的基本理论。这些成果作为论文集发表。2006年，我校邀请了Mori Izuru教授来我校主讲非交换代数几何，介绍了非交换环理论的一个重要领域——非交换代数几何的发展历史和国外的发展趋势。我们还邀请了井usa清（美国布兰迪斯大学）和姚慕尚教授（中国复旦大学）。通过与许多数学家的这些研究，我们完成了一些二次型的研究，并得到了表征为对应于代数的一个Cartan矩阵的二次型。我们还用代数的卡坦矩阵的柯赛特矩阵的周期性来描述这些条件。结果证明了Coxeter矩阵的特征根是代数整数，这在这些研究中具有重要意义。

In 2004, we attended two international conference, the fourth Japan-China-Korea Iternational symposium on ting theory held in Nanjing, China from July 24 to 26 and the 11th International conference on representation theory of algebras to introduce the recent results and observe the trend of study.In 2005, we held the 38th ring and representation theory symposium in Aichi Institute of Technology and we invited professor Raphael Rouquier (University Denis Diderot-Paris 7) who study closely related us and asked him lectures about elementary theory for solving Broue conjecture. These achievement are published as the proceedings.In 2006, we invited professor Izuru Mori to ask lectures about non-commutative algebraic geometry, which is expected an important field of non-commutative ring theory, including its history and the trend in abroad. Also we invited Kiyoshi Igusa (Brandise, U. S. A) and Professor Yao Mushung (Fudan, China).Through these research with many mathematicians, we completed the research of some kind of quadratic forms and we get the characterization to be the quadratic form corresponding to a Cartan matrix of an algebra. Also we characterized these conditions by the periodicity of a Coxeter matrix of a Cartan matrix of an algebra. As the result, we showed that the characteristic root of a Coxeter matrix is algebraic integer and this plays important role in these study.