Polynomial identities of RA2 loop algebras (Q1283650)

Let \(L\) be a loop, and let \(R\) be a ring. Then the loop ring \(RL\) is the free \(R\)-module with basis \(L\) and multiplication induced by the binary operation of \(L\). If \(R\) is a field then \(RL\) is a loop algebra. In the sequel, let \(R=K\) be a field of characteristic \(\neq 2,3\). A loop whose loop algebra satisfies the alternative identities is called an RA loop, and an RA2 loop is a loop whose loop algebra over a field of characteristic 2 is alternative but not associative. There are three RA2 loops \(L_3\), \(L_4\), and \(L_5\) of order 16 which are not RA loops. Their loop algebras \(KL_i\), \(i=1,2, 3\), over a field \(K\) of characteristic \(\neq 2\) may be written in the form \(KL_i \simeq 8K\oplus{\mathcal A}_i\), where \({\mathcal A}_i\) is a simple algebra of dimension eight over \(K\). The algebras \({\mathcal A}_i\) are described by their multiplication tables. The authors study the polynomial identities and the polynomial central identities of the loop algebras of the three loops \(L_3\), \(L_4\), and \(L_5\). They apply a modified version of the Cayley-Dickson process to the algebras \({\mathcal A}_3\), \({\mathcal A}_4\), and \({\mathcal A}_5\) to obtain the polynomial identities of small degree of these algebras. It turns out that the algebras \({\mathcal A}_i\), \(i=3, 4, 5\), do not satisfy any polynomial identity or central identity of degree \(\leq 3\). The polynomial identities of degree 4 of \({\mathcal A}_i=3, 4, 5\), are consequences of ten identities, and the degree 4 polynomial central identities of \({\mathcal A}_i\), \(i=3,4,5\), are consequences of two more identities. Moreover, \(f\) is a polynomial identity (central identity) of \({\mathcal A}_i\), \(i= 3, 4, 5\) if and only if \(f\) is a polynomial identity (central identity) of \(KL_i\). Thus the authors have found all polynomial identities of degree \(\leq 4\) of \(KL_i\).