REPRESENTATIONS OF RELATIVELY FREE ALGEBRAS AND CANONIZATION THEOREMS

Канель-Белов, А.Я.

Автор: Канель-Белов А.Я.
Сборник: Алгебра, теория чисел, дискретная геометрия и многомасштабное моделирование: современные проблемы, приложения и проблемы истории Материалы XXII Международной конференции, посвященной 120-летию со дня рождения академика А. Н. Колмогорова и 60-летию со дня
Тезисы
Год издания: 2023
Место издания: Тульский государственный педагогичекий университет им. Л. Н. Толстого, 2023 Тула Тула
Последняя страница: 96
Номер статьи: 97
Аннотация: Representations of relatively free algebrasand canonization theoremsA. Kanel-Belov (Israel, Ramat Gan)Bar-Ilan Universitye-mail: kanelster@gmail.comThe proof of local nite basis property and local representability of relatively free algebras wasdone, according to the Kemer programm. In order to make the proof more acceptable, the followingquestion arises. What does it provide for the community? Why should people read it?Joint work of the speaker with Louis Rowen and Uzi Vishne discovered relations withnoncommutative algebraic geometry and provided new insights for representation theory.Consider representation 𝜌 of 𝑘-algebra 𝐴 to matrix algebra over algebraic closed eld 𝐾, whichis an ane space. Then Zarissky closer of 𝜌(𝐴) satises the same identities and its natural toinvestigate representations up to Zarissky closure which usually is not a linear span if 𝑘 is nite. 𝜌(𝐴)decomposes into a sum of prime components and Pierce components of radical. First canonizationtheorem says that it can be reduced to upper block-triangular case. Blocks are either glued (may beup to Frobenius twist) or independant. Second canonization theorem provides information for quiveror pseudoquiver. Third canonisation theorem says about quiver transformation under factoringby representable 𝑇-ideal. And Fourth (projective) canonization theorem provide existence of nonidentities of some canonical structure inside non-zero 𝑇-ideal and Phoenix property (hiking) sayingthat any element of 𝑇-ideal generated by hiked polynomial can be restituted to the same form.Because one can provide via substitutions on hiked polynomial structure of the Notherean module,Specht properties and local representability follows.
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