Nilpotent symplectic alternating algebras II

Abstract

In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension 10 over any field. It is known that symplectic alternating algebras over GF(3) correspond to a special rich class [Formula: see text] of 2-Engel 3-groups of exponent 27 and under this correspondence we will see that the nilpotent algebras correspond to a subclass of [Formula: see text] that are those groups in [Formula: see text] that have an extra group theoretical property that we refer to as being powerfully nilpotent and can be described also in the context of [Formula: see text]-groups where [Formula: see text] is an arbitrary prime.

Nilpotent symplectic alternating algebras III

“…Symplectic alternating algebras originate from a study of powerful 2-Engel groups [1], [5] and there is a 1-1 correspondence between a certain rich class of powerful 2-Engel 3-groups of exponent 27 and SAAs over the field GF (3). We refer to [2] and [6] for a more detailed discussion of this as well as some general background to SAAs.…”

“…In this paper we finish our classification of nilpotent symplectic alternating algebras of dimension 10 that we started in [3]. Our approach like there relies on a general theory that we developed in [2].…”

“…, n − 2, and x 1 y 2 , y 1 y 2 are linearly independent ( see [2] Theorem 3.4). In [3] we found all the nilpotent algebras with a non-isotropic centre and we gave a full classification of the nilpotent SAAs with an isotropic centre of dimension 5 and 3. Here we deal with the two remaining cases, namely when the centre has dimension 2 or 4.…”

“…3 and U L 2 is 1-dimensional. The algebra can be given by a presentation of the form P(2,5) 10 (r) : (x 2 y 3 , y 5 ) = r, (x 3 y 4 , y 5 ) = 1, (x 1 y 2 , y 5 ) = 1, (y 1 y 2 , y 3 ) = 1,…”

Nilpotent symplectic alternating algebras I

“…Symplectic alternating algebras originate from a study of powerful 2-Engel groups [1], [5] and there is a 1-1 correspondence between a certain rich class of powerful 2-Engel 3-groups of exponent 27 and SAAs over the field GF (3). We refer to [2], [4] and [6] for more detailed discussion of this as well as some general background to SAAs.…”