Quadratic algorithm to compute the Dynkin type of a positive definite quasi-Cartan matrix,Mathematics of Computation

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Quadratic algorithm to compute the Dynkin type of a positive definite quasi-Cartan matrix
Mathematics of Computation ( IF 2 ) Pub Date : 2020-08-01 , DOI: 10.1090/mcom/3559
Bartosz Makuracki , Andrzej Mróz

Abstract:Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite quasi-Cartan matrix $A\in \mathbb{M}_n(\mathbb{Z})$ is $\mathbb{Z}$ -equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of

. We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of

, of the pessimistic arithmetic (word) complexity $\mathcal {O}(n^2)$ , significantly improving the existing algorithms. As an application we note that our algorithm can be used as a positive definiteness test for an arbitrary quasi-Cartan matrix, more efficient than standard tests. Moreover, we apply the algorithm to study a class of (symmetric and non-symmetric) quasi-Cartan matrices related to Nakayama algebras.

中文翻译：

二次算法，用于计算正定准Cartan矩阵的Dynkin类型

摘要：Cartan矩阵和拟Cartan矩阵在Lie理论，表示论和代数图论等领域起着重要作用。已知的是，每一个（连接）正定准嘉当基质是换算用Dynkin图的嘉当矩阵，称为Dynkin型的。我们提出了一种符号图论算法来计算Dynkin类型的，其悲观算术（单词）复杂度 $A \ in \ mathbb {M} _n（\ mathbb {Z}）$ $\ mathbb {Z}$

$\ mathcal {O}（n ^ 2）$ ，大大改善了现有算法。作为应用，我们注意到我们的算法可以用作任意准Cartan矩阵的正定性检验，比标准检验更有效。此外，我们将算法应用于研究与中山代数有关的一类（对称和非对称）拟Cartan矩阵。

更新日期：2020-08-01

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