Graded Identities of Some Simple Lie Superalgebras

Abstract

We study Z 2 -graded identities of Lie superalgebras of the type b(t), t ≥ 2, over a field of characteristic zero. Our main result is that the n-th codimension is strictly less than (dim b(t)) n asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra b(t), t ≥ 3.

Graded PI-exponents of simple Lie superalgebras

“…by Lemma 2.1 and the inequality (11). Now it follows from (12) that c gr n (L) > 1 n 2d 2 +2d+1 1 (2n 4 0 ϕ(n 0 )) q (a−ε) qn0 .…”

“…for all n=1, 2, ... as it was mentioned in [11]. Note also that m λ,µ =0 in (3) only if λ⊢k, µ⊢n−k are partitions with at most d components, that is λ=(λ 1 , ..., λ p ), µ= (µ 1 , ..., µ q ) and p, q≤d=dim L.…”

Identities of graded simple algebras

“…An existence of the Z 2 -graded PI-exponent for any finite dimensional simple Lie superalgebra has recently been proved in [13]. Note that for finite dimensional Lie superalgebras, both graded and ordinary PI-exponents can be fractional [11,14,15]. The main purpose of this paper is to prove the existence of graded PI-exponents for any finite dimensional graded simple algebra (see Theorem 2).…”

Pauli gradings on Lie superalgebras and graded codimension growth

“…All remaining components L k , k = 0, ±1, are zero. Clearly, P n1,...,n d (L) = 0 only if (20) |n 1 + · · · + n a − n a+1 · · · − n a+b | ≤ 1…”

“…Our main result is Theorem 1 below, stating that exp G (P (t)) = t 2 −1+t √ t 2 − 1. Note that Theorem 1 is true for t = 2 although P (2) is not simple and exp G (P (2)) = 3 + 2 √ 3 holds for both Pauli grading and the canonical Z 2 -grading (see [20]).…”