# Structure of Degenerate Block Algebras

Structure of Degenerate Block Algebras Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map $$\phi$$ : A $$\times$$ A $$\rightarrow$$ F, we define a Lie algebra $${\cal L}$$ = $${\cal L}$$ (A, $$\phi$$ ) over F with basis {e x | x $$\in$$ A/{0}} and Lie product [e x,e y] = $$\phi$$ (x,y)e x+y. We show that $${\cal L}$$ is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der $${\cal L}$$ of $${\cal L}$$ is a complete Lie algebra. We describe the double extension D( $${\cal L}$$ , T) of $${\cal L}$$ by T, where T is spanned by the locally finite derivations of $${\cal L}$$ , and determine the second cohomology group H 2(D( $${\cal L}$$ , T),F) using anti-derivations related to the form on D( $${\cal L}$$ , T). Finally, we compute the second Leibniz cohomology groups HL 2( $${\cal L}$$ , F) and HL 2(D( $${\cal L}$$ , T), F). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algebra Colloquium Springer Journals

# Structure of Degenerate Block Algebras

, Volume 10 (1) – Jan 1, 2003
10 pages

/lp/springer-journals/structure-of-degenerate-block-algebras-9q3SVqxKal
Publisher
Springer Journals
Subject
Mathematics; Algebra; Algebraic Geometry
ISSN
1005-3867
eISSN
0219-1733
DOI
10.1007/s100110300007
Publisher site
See Article on Publisher Site

### Abstract

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map $$\phi$$ : A $$\times$$ A $$\rightarrow$$ F, we define a Lie algebra $${\cal L}$$ = $${\cal L}$$ (A, $$\phi$$ ) over F with basis {e x | x $$\in$$ A/{0}} and Lie product [e x,e y] = $$\phi$$ (x,y)e x+y. We show that $${\cal L}$$ is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der $${\cal L}$$ of $${\cal L}$$ is a complete Lie algebra. We describe the double extension D( $${\cal L}$$ , T) of $${\cal L}$$ by T, where T is spanned by the locally finite derivations of $${\cal L}$$ , and determine the second cohomology group H 2(D( $${\cal L}$$ , T),F) using anti-derivations related to the form on D( $${\cal L}$$ , T). Finally, we compute the second Leibniz cohomology groups HL 2( $${\cal L}$$ , F) and HL 2(D( $${\cal L}$$ , T), F).

### Journal

Algebra ColloquiumSpringer Journals

Published: Jan 1, 2003