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Quadratic algorithm to compute the Dynkin type of a positive definite quasi-Cartan matrix
Mathematics of Computation ( IF 2.2 ) Pub Date : 2020-08-01 , DOI: 10.1090/mcom/3559 Bartosz Makuracki , Andrzej Mróz
Mathematics of Computation ( IF 2.2 ) 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 is -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 , 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 ,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
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 ,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。