On Degenerations of Algebras over an Arbitrary Field

Abstract

We investigate degenerations of $n$-dimensional algebras over an arbitrary infinite field paying particular attention to algebras satisfying the identity $[x,x]=0$ (where $[x_1,x_2]$ denotes the product of the elements $x_1,$ $x_2$ of the algebra). We show, for $n\ge3$, that there are precisely two non-isomorphic $n$-dimensional algebras which satisfy the above identity and, in addition, have the $n$-dimensional abelian Lie algebra $\mathfrak{a}_n$ as their only proper degeneration. These are the algebras $\mathfrak{h}_n=\mathfrak{h}_3\oplus\mathfrak{a}_{n-3}$, where $\mathfrak{h}_3$ is the Heisenberg algebra, and $\mathfrak{r}_n$ which is, up to isomorphism, the only $n$-dimensional algebra other than $\mathfrak{a}_n$ that satisfies both the identity $[x,x]=0$ and the condition that the product of any two of its elements is a linear combination of the same two elements (in particular, $\mathfrak{h}_n$ and $\mathfrak{r}_n$ are both Lie algebras).