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On the Bergman's Centralizer Theorem by Quantizationдипломная работа (Магистр)
- Научный руководитель: Kanel-Belov Alexey
- Автор: Wenchao Zhang
- Тип: Магистр
- Организация, в которой проходила защита: Bar-Ilan University
- Год защиты: 2018
- Аннотация: This thesis is dedicated to the study of subalgebras of free associative algebras and the quantization method for free associative algebras. We found a new proof of the first step of Bergman's centralizer theorem by using Kontsevich's quantization method, namely that any coomutative subalgebra have transcendentence degree 1. Our new proof doesn't not directly work on the free associative algebra, but the algebra of generic matrices related to this algebra. We gave some background on the matrix representations of algebras and a necessary theorem by Amitsur and Levitzki. Meanwhile, we found a canonical homomorphism from the free associative algebra to the algebra of generic matrices, and proved that if there is a commutative subalgebras of transcendence degree two in the free associative algebra, then there also exists a commutative subalgebra of transcendence degree two in the algebras of generic matrices. After that, we discussed the quantization on the generic matrices. If there exist two commuting generic matrices which are algebraically independent, then after a proper extension of $k[x_{ij}^{(\nu)}]$ and a proper choice of basis, we can diagonalize these two generic matrices. Then, we computed the quantization commutator of these two matrices which gives us nonzero. And this leads to a contradiction since canonical quantization sending zero element in the algebra of generic matrices to zero. Hence, we obtained the result that there is no commuta\-tive subalgebras of transcendence degree two. Since the centralizer subalgebra $C(a;k\langle X\rangle)$ is commutative hence of transcendence one.
- Добавил в систему: Белов Алексей Яковлевич
Прикрепленные файлы
| № | Имя | Описание | Имя файла | Размер | Добавлен |
|---|---|---|---|---|---|
| 1. | Сертификат | WENCHAO-MSc.pdf | 188,7 КБ | 28 ноября 2018 [AlexeiBelov] | |
| 2. | Полный текст | Текст диплома | mthesis_istina_msu_ru.pdf | 255,0 КБ | 29 октября 2018 [AlexeiBelov] |