Sesión Álgebra, Álgebra Conmutativa, Teoría de GruposPolynomial identities and codimensions for Witt algebras in characteristic 2
Airton Muniz Cordeiro
Universidade Estadual de Campinas, Brasil - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
Let \(U_1\) and \(W_1\) denote the Lie algebras of derivations of the Laurent polynomial algebra \(K[t,t^{-1}]\) and the polynomial ring \(K[t]\), respectively. Both algebras are naturally \(\mathbb{Z}_i\)-graded. Although these structures have been studied since the twentieth century and have important applications in physics, concrete results concerning their graded identities have only been obtained recently. In particular, Freitas, Koshlukov, and Krasilnikov (2015) determined a basis for the \(\mathbb{Z}\)-graded identities of \(W_1\)in characteristic zero, while Fideles and Koshlukov in (2022) and (2023) determined the \(\mathbb{Z}\)-graded identities of \(U_1\) and \(W_1\) in positive characteristic. In this work, we investigate the \(\mathbb{Z}_i\)-graded identities of these algebras over a field of characteristic 2. As a consequence, we obtain a basis for their ordinary identities in characteristic 2, thereby partially solving a long-standing open problem, and we calculate their codimension sequences. Furthermore, we show that these algebras generate the variety of metabelian Lie algebras in characteristic 2.
Trabajo en conjunto con: Claudemir Fideles (Universidade Estadual de Campinas).
Referencias
[1] Fideles, Claudemir and Koshlukov, Plamen. $\mathbb{Z}$-graded identities of the Lie algebras $U_1$ in characteristic 2. Mathematical Proceedings of the Cambridge Philosophical Society, v. 174, p. 49–58, 2022.
[2] Fideles, Claudemir and Koshlukov, Plamen. $\mathbb{Z}$-graded identities of the Lie algebras $U_1$. Journal of Algebra, v. 633, p. 668–695, 2023.
[3] Freitas, José A. Koshlukov, Plamen and Krasilnikov, Alexei. $\mathbb{Z}$-graded identities of the Lie algebra $W_1$. Journal of Algebra, v. 427, p. 226–251, 2015.