In this paper nilpotent six-dimensional Lie algebras are full classified over arbitrary fields. The history of this classification problem is long, beginning in 1891 in [K. A. Umlauf, Über die Zusammensetzung der endlichen continuirlichen Transformationsgruppen, insbesondere der Gruppen vom Range Null. Ph.D. thesis, University of Leipzig, Germany (1891; JFM 24.0333.04)]. Since then, several classifications have appeared, for instance over fields of characteristic zero [V. V. Morozov, Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No. 4(5), 161–171 (1958; Zbl 0198.05501)], and more recently over algebraically closed fields and over fields of characteristic not 2. This paper completes the classification, giving an unified treatment for all the ground fields. A complete list of the isomorphism classes of these algebras is exhibited, with their multiplication tables enclosed. Concretely, there are \(26+4s\) classes if the field \(\mathbb{F}\) has characteristic not 2, and \(30+2s+4t\) if the characteristic of the field is 2, where \(s\) is the index of \((\mathbb{F}^*)^2\) in \(\mathbb{F}^*\) and \(t\) is the number of equivalence classes in \(\mathbb{F}^*\) of the relation \(\sim\) given by \(\alpha\sim\beta\) when there is \(\gamma\in\mathbb{F}^*\) and \(\delta\in\mathbb{F}\) such that \(\alpha=\gamma^2\beta+\delta^2\). Thus the number of isomorphy classes can be infinite, in contrast with the number of isomorphy classes of nilpotent Lie algebras of dimension up to 5 (one of dimension 2, two of dimension 3, three of dimension 4 and 9 of dimension 5), which is considerably smaller even for algebraically closed fields or perfect fields.
The idea of the used methodology is to obtain the Lie algebras as central extensions of Lie algebras of smaller dimension. Each finite-dimensional nilpotent Lie algebra \(L\) is either \(K\oplus \mathbb{F}\) with \(K\) an ideal, or isomorphic to \(K/Z(K)\) with \(Z(K)\subset[K,K]\). In this latter case \(K\) is called a descendant of \(L\). The isomorphism types of the descendants of \(L\) are in one-to-one correspondence with the \(\text{\text {Aut}}(L)\)-orbits of the second cohomological space \(H^2(L,V)=Z^2(L,V)/B^2(L,V)\). The cocycles \( \vartheta\in Z^2(L,V)\) are the alternating bilinear maps \(\vartheta: L\times L\to V\) such that \(\vartheta([x_1,x_2],x_3)+\vartheta([x_2,x_3],x_1)+\vartheta([x_3,x_1],x_2)=0\) for all \(x_i\in L\), and the coboundaries are the maps \(\eta_\nu: L\times L\to V\) defined as \(\eta_\nu(x,y)=\nu([x,y])\) for \(\nu: L\to V\) a linear map. Now, each \(\vartheta\in Z^2(L,V)\) defines a Lie algebra \(L_\vartheta=L\oplus V\) by \([x+v,y+w]:=[x,y]+\vartheta(x,y)\), which is a central extension of \(L\) isomorphic to \(L_{\vartheta+\eta_\nu}\) (and conversely). The algebra \(L_\vartheta\) is a descendant of \(L\) when \(Z(L)\) does not intersect the radical of \(\vartheta\). For two cocycles \(\vartheta\) and \(\eta\) such that \(L_\vartheta\) and \(L_\eta\) are descendants of \(L\), these algebras are isomorphic if and only if there is \(\phi\in\operatorname{Aut}(L)\) such that \(\langle\phi(\vartheta_1),\dots,\phi(\vartheta_s)\rangle\) is the same subspace of \(H^2(L,V)\) as \(\langle\eta_1,\dots,\eta_s\rangle\), where a basis \(\{e_1,\dots,e_s\}\) of \(V\) has been chosen and the cocycles have been written as \(\vartheta(x,y)=\sum\vartheta_i(x,y)e_i\) and \(\eta(x,y)=\sum\eta_i(x,y)e_i\). For the concrete computations of all such orbits, the authors have used some geometric invariants, such as the Gram determinant or the Arf invariant, more appropriate in the characteristic 2 case.
A byproduct of this work is the determination, for an arbitrary vector space \(V\) of dimension 4, of the \({\text{GL}}(V)\)-orbits on the two-dimensional subspaces of \(V\wedge V\).