Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ Dessislava H. Kochloukova 14 February 2006 2006 2 14 109 127 721-5093-Kochloukova-2005 10.wxyz/C2005V72P1p109 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5093-Kochloukova/ We classify the Hopf algebras $U(L) \# kQ$ of homological type $FP_2$ where $L$ is a Lie algebra and $Q$ an Abelian group such that $L$ has an Abelian ideal $A$ invariant under the $Q$-action via conjugation and $U(L/A) \# kQ$ is commutative. This generalises the classification of finitely presented metabelian Lie algebras given by J. Groves and R. Bryant. 16S40, 57T05 MR2162297 02212189 Partially supported by ``bolsa de produtividade de pesquisa" from CNPq, Brazil. G. Baumslag J. Pure Appl. Algebra MR437608 G. Baumslag; Some remarks on finitely presented algebras, \textit{J. Pure Appl. Algebra} \textbf{8} (1976), pp.~187--196. M. Bestvina and N. Brady Invent. Math. MR1465330 M. Bestvina and N. Brady; Morse theory and finiteness properties of groups, \textit{Invent. Math.} \textbf{129} (1997), pp.~445--470. R. Bieri and J.R.J. Groves Proc. London Math. Soc. (3) MR670042 R. Bieri and J.R.J. Groves; Metabelian groups of type $FP{\infty }$ are virtually of type $FP$, \textit{Proc. London Math. Soc. (3)} \textbf{45} (1982), pp.~365--384. R. Bieri and R. Strebel Proc. London Math. Soc. (3) MR591649 R. Bieri and R. Strebel; Valuations and finitely presented metabelian groups, \textit{Proc. London Math. Soc. (3)} \textbf{41} (1980), pp.~439--464. R. Bryant and J.R.J. Groves J. London Math. Soc. (2) MR1721814 R. Bryant and J.R.J. Groves; Finite presentation of abelian-by-finite dimensional Lie algebras, \textit{J. London Math. Soc. (2)} \textbf{60} (1999), pp.~45--57. R. Bryant and J.R.J. Groves J. Algebra MR1704674 R. Bryant and J.R.J. Groves; Finitely presented Lie algebras, \textit{J. Algebra} \textbf{218} (1999), pp.~1--25. D.H. Kochloukova Comm. Algebra D.H. Kochloukova; Finite presentability and the homological type $FPm$ for a class of Hopf algebras, \textit{Comm. Algebra} (to appear). S. Montgomery CBMS Regional Conference Series in Mathematics 82 American Mathematical Society MR1243637 S. Montgomery; \textit{Hopf algebras and their actions on rings}, CBMS Regional Conference Series in Mathematics 82 (American Mathematical Society, Providence, R.I., 1993).