The Grassmann Algebra and its Differential Identities

Growth of differential identities

“…Since the ordinary polynomial identities and corresponding codimensions are obtained by let L acting on A trivially (or L is the trivial Lie algebra), the algebra U T 2 of 2 × 2 upper triangular matrices regarded as L-algebra where L acts trivially on it generates an L-variety of almost polynomial growth (see [8,4]). Clearly another example of algebras generating an L-variety of almost polynomial growth is the infinite dimensional Grassmann algebra G where L acts trivially on it (see [8,13]). Notice that in the ordinary case Kemer in [8] proved that U T 2 and G are the only algebras generating varieties of almost polynomial growth.…”

“…If we evaluate x = βe 11 + e 12 + e 22 , where β ∈ F , β = 0, and y = e 11 , then we get (13) q i=0 (−1) p−1 β i α i = 0.…”

“…, β q+1 ∈ F , where β j = 0, β j = β k , for all 1 ≤ j = k ≤ q + 1. Then from (13) we obtain a homogeneous linear system of q + 1 equations in the q + 1 variables α i , i = 0, . .…”