ИСТИНА

\begin{document} \renewcommand{\thefootnote}{\fnsymbol{footnote}} %\footnotetext{\emph{2010 Mathematics Subject Classification:} 14R10, 14R20} \renewcommand{\thefootnote}{\arabic{footnote}} \fontsize{12}{12pt}\selectfont \title{\bf Canonization theorems, Specht Problem and Representation Theory} \renewcommand\Affilfont{\itshape\small} \author[1]{Alexei Kanel-Belov\thanks{kanel@mccme.ru}} \affil[1]{Mathematics Department, Bar-Ilan University, Ramat-Gan, 52900, Israel} \date{} \maketitle \renewcommand{\abstractname}{Abstract} \begin{abstract} The general Specht problem says "does given set of identities of associative algebras stabilize? i.e. does any set of identities can be deduced from finite subset? Specht kept in mind case of field of characteristic zero, and this problem was solved by A.R.Kemer in affirmative way. In positive characteristics for case of finite number of variables (``Local Specht problem'') A.Belov gave an affirmative answer and in general counterexamples was constructed. Solution of Local Specht problem over an arbitrary field (including finite) provided some new point of view on ring representation. Let $A$ be $PI$-algebra over $k$, $\rho: A\to M_n(K)$ be its representation, $K$ is algebraically closed. $M_n(K)$ can be treated as and $n^2$ dimensional affine space over $K$ and polynomial identities of $\rho(A)$ and its Zarissky closure coincide. We came to interesting problem of classification of representations up to Zarissky closure. Zarrissky closed algebras have Pierce decomposition and radical part decomposition. {\bf First canonization theorem} says about block-diagonal structure of semisimple part (see Kanel-Belov A., Rowen Louis H.; Vishne, Uzi, ``Structure of Zariski-closed algebras.'', Trans. Amer. Math. Soc., 362:9 (2010), 4695--4734 , arXiv: 1109.4912 for details). {\bf Second canonization theorem} says how to construct a quiver in canonical way (Kanel-Belov A., Rowen Louis H.; Vishne, Uzi, ``Full quivers of representations of algebras'', Trans. Amer. Math. Soc., 364:10 (2012), 5525--5569 , arXiv: 1109.4916, Kanel-Belov A., Rowen Louis H.; Vishne, Uzi, ``Application of full quivers of representations of algebras, to polynomial identities.'', Comm. in Algebra, 39 (2011), 4536--4551). {\bf Third canonization theorem} says about how factorizing on a representable $T$-ideal, closed under coefficient ring multiplication produce quiver reduction branching process (Alexei Belov-Kanel, Louis H. Rowen and Uzi Vishne, ``PI-varieties associated to full quivers of representations of algebras'', Trans. Amer. Math. Soc., 365:5 (2013), 2681--2722). And {\bf Fourth canonization theorem A} (of ``Hiking'') says about canonical non-identities of $A$, {\bf Fourth canonization theorem B} says about canonical non-identities of inside arbitrary characteristic ideals of $A$. (Alexei Belov-Kanel, Louis Rowen, Uzi Vishne, ``Specht's problem for associative affine algebras over commutative Noetherian rings'', Trans. Amer. Math. Soc., 367:8 (2015), 5553--5596 , arXiv: 1308.3055, Alexei Belov-Kanel, Louis Rowen, Uzi Vishne, ``Representability of affine algebras over an arbitrary field'', 2018 (Published online) , 20 pp., arXiv: 1805.04450 Specht problem follows from these theorems immediately. Joint work with Louis Rowev and Uzi Vishne The talk was supported by Russian Science Foundation grant N 17-11-01377 \end{abstract} \emph{Key words:} $PI$-algebra, Specht Problem, Relativelly Free Algebra, Representations. \end{document}