Identities for a parametric Weyl algebra over a ring

Identities for subspaces of the Weyl algebra

“…Considering i = 0, j = 1 and i = 1, j = 0 in equality (10) we obtain that 4α 1 + 2α 2 + α 3 = 0 and α 4 + 2α 5 + 4α 6 = 0, (12) respectively. Let p = 2.…”

“…Similarly, equality (10) implies that ij(α 1 +α 2 +α 5 +α 6 )+jα 3 +iα 4 = 0 for all i, j ≥ 0 with i+j ≥ 1. Equalities (12) imply that α 3 = α 4 = 0. Applying equality (8) we can see that…”

“…The situation is drastically different in case p > 0. Namely, A 1 is PI-equivalent to the algebra M p of all p × p matrices over F. Moreover, the Weyl algebra A 1 over an arbitrary associative (but possible non-commutative) F-algebra B is PI-equivalent to the algebra M p (B) of all p × p matrices over B (see Theorem 4.9 of [12] for more general result).…”

Weak polynomial identities of small degree for the Weyl algebra

“…The situation is drastically different in case p > 0. Namely, A 1 is PI-equivalent to the algebra M p of all p × p matrices over F. Moreover, the Weyl algebra A 1 over an arbitrary associative (but possible non-commutative) F-algebra B is PI-equivalent to the algebra M p (B) of all p × p matrices over B (see Theorem 4.9 of [19] for more general result). Polynomial identities for A (−,s) 1 and other subspaces of A 1 were studied in [20,21].…”