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

2004年7月24日至26日在中国南京召开的第四届日中韩代数表示理论国际研讨会和第十一届代数表示理论国际会议，介绍了近年来的研究成果，并观察了研究趋势。我们在爱知工业大学举办了第38届环与表示论研讨会，邀请了Raphael Rouquier教授（大学丹尼斯狄德罗-巴黎7）谁研究密切相关的我们，并要求他讲座有关的基本理论解决布鲁猜想。2006年，我们邀请了Izuru Mori教授就非交换环理论的一个重要领域--非交换代数几何进行了讲座，包括它的历史和国外的发展趋势。我们还邀请了Kiyoshi Igusa（Brandise，U。S. A）和姚木雄教授（复旦大学），通过与众多数学家的合作，我们完成了对一类二次型的研究，得到了对应于代数的Cartan矩阵的二次型的特征。我们还利用代数的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.