Weak polynomial identities for the matrix algebra of order two
The weak polynomial identities for the Lie algebra \(sl_2\) of \(2\times 2\) traceless matrices are studied in the case when the base field is infinite of positive characteristic different from 2. Recall that an associative polynomial is a weak identity for \(sl_2\) if it vanishes under any substitution of the variables by elements of \(sl_2\). It is proved that all weak identities for \(sl_2\) follow from the single identity \([x\circ y,z]=0\), where \(a\circ b=(ab+ba)/2\), and \([a,b]=ab-ba\). In characteristic zero this theorem is due to \textit{Yu. P. Razmyslov} [Algebra Logika 12, 83-113 (1973; Zbl 0282.17003)]. The proof uses methods of \textit{C. de Concini} and \textit{C. Procesi} [Adv. Math 21, 330-354 (1976; Zbl 0347.20025)], which were developed further by \textit{S. Yu. Vasilovskij} [Algebra Logika 28, No. 5, 534-554 (1989; Zbl 0702.17013)].
scientific article; zbMATH DE number 1013941
scientific article