Quantization of Galois theory

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610004
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: Umemura Hiroshi 名古屋大学, 多元数理科学研究科, 名誉教授 (40022678)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: Picard-Vessiot 理論 / 量子ガロア理論 / Hopf 代数

Outline of Final Research Achievements

We can interpret differential Galois theory as a theory of Module algebras under the Hopf algebra of the co-orrdinate ring of the 1 dimensional additive group. Hopf algebraists succeeded in generalizing the theory of linear equations from this view point, to a co-commutative Hopf algebra acting on a commutative module algebra. There arises a natural question whether we can extend this general theory to non-commutative case, or we can quantize it. Using the notion of Galois hull born in our study of Galois theory of non-linear differential equations, we successfully established a general Galois theory of non-commutative linear equations with constant coefficients.

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