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

2004年，我们参加了在南京举行的第四届日本-中国-韩国国际学术研讨会，中国于7月24-26日参加了第11届国际代数表示理论会议，介绍了最近的研究成果并观察了研究趋势。2005年，我们在爱知理工学院召开了第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(Denis Diderot-Paris 7)，为他讲授解决Broue猜想的初等理论。2006年，我们邀请了Izuru Mori教授讲授非对易代数几何，这是非对易环论的一个重要领域，包括它的历史和国外的发展趋势。我们还邀请了井上清(美国布兰迪斯)和姚木松教授(复旦，中国)，通过与许多数学家的研究，我们完成了对某种二次型的研究，得到了与代数的Cartan矩阵相对应的二次型的刻划。我们还用代数的Cartan矩阵的Coxeter矩阵的周期性刻画了这些条件。结果表明，Coxeter矩阵的特征根是代数整数，这在这些研究中起着重要的作用。

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.