On the Structure of Bernstein Algebras

On nilpotent evolution algebras

“…If dim(E) = 4, then dim(ann(E)) = 1 by Corollary 2.6. The possible types are (ordered lexicographically) [1,3], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1], and the result follows from Theorems 4.7 and 4.11.…”

“…• If the type is [2,1,2], there is a natural basis {x, y, a, u, v} with ann(E) = span {u, v}, ann 2 (E) = span {a, u, v}. As x 2 ∈ ann 2 (E) \ ann(E), {x, y, x 2 , u, v} is another natural basis.…”

“…This condition already implies E to be indecomposable. The possible types of E, ordered lexicographically, are [1,4] We will classify first those algebras isomorphic to the algebras in Theorems 4.7 and 4.11.…”

“…(iv) If the type is [1,1,1,2], then E is isomorphic to an algebra with one of the following graphs , , γ , γ β .…”

“…Any algebra of type [1,1,3] is isomorphic to an algebra E(U, b, g), with dim(U) = 3, as in Theorem 4.7(ii), and we may change the symmetric endomorphism g by µg + νid for µ, ν ∈ F, µ = 0.…”