Weak polynomial identities of small degree for the Weyl algebra (Q6615600)

The authors study weak polynomial identities for the Weyl algebra \( A_1 \) (an associative algebra over a field \( \mathbb{F} \) with unity and generators \( x \), \( y \) that satisfy the defining relation \( yx = xy + 1 \)) over an infinite field \( \mathbb{F} \) of arbitrary characteristic. They do not find an explicit basis for these weak identities, but they show weak identities of minimal degree and present, in the form of a conjecture, a candidate for a generating set of the basis.\N\NThe motivation and background stem from the significance of studying weak identities, initially developed by Razmyslov, and the applications of this theory in finding ordinary identities and central polynomials.\N\NLet \((A_1, V)\) be the pair consisting of the Weyl algebra \( A_1 \) and the vector space \( V = \text{span}_{\mathbb{F}} \{ x, y \} \).\N\NThe main results are:\N\begin{itemize}\N\item \NThe minimal degree of a non-trivial weak polynomial identity for the pair \((A_1, V)\) is three. Moreover, any weak identity of \((A_1, V)\) of degree three lies in the \( L \)-ideal generated by the polynomials \( \Gamma_3 \) and \( \mathrm{St}_3 \);\N\item \Nany weak polynomial identity for the pair \((A_1, V)\) of degree 4 or 5 is a weak consequence of the polynomials \( \Gamma_3 \), \( \mathrm{St}_3 \), and \( T_4 \);\N\end{itemize}\Nwhere \( \Gamma_3 \), \( \mathrm{St}_3 \), and \( T_4 \) are the polynomials:\N\begin{align*}\N\Gamma_3 (x_1, x_2, x_3) & = \left[ [x_1, x_2], x_3 \right], \\\N\mathrm{St}_3 (x_1, x_2, x_3) &= x_1 [x_2, x_3] - x_2 [x_1, x_3] + x_3 [x_1, x_2], \\\NT_4(x_1, \ldots, x_4) &= [x_1, x_2][x_3, x_4] - [x_1, x_3][x_2, x_4] + [x_2, x_3][x_1, x_4].\N\end{align*}\NWith these results, the authors conjecture that the ideal of all weak polynomial identities for the pair \((A_1, V)\) is generated by the polynomials \( \Gamma_3 \), \( \mathrm{St}_3 \), and \( T_4 \).\N\NTo develop their proofs, the authors define semi-reduced, reduced, and completely reduced bracket-monomials. These concepts turn out to be fundamental in the proofs, following a line of reasoning that resembles some steps of the PBW theorem proof, though in a much simpler form.