# New PDF release: A first course of homological algebra

By D. G. Northcott

ISBN-10: 0521299764

ISBN-13: 9780521299763

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Additional resources for A first course of homological algebra

Example text

The proof is by induction on dim L. If dim L = 0, then the statement is trivial. So suppose t h a t dim L > 1. By induction we m a y suppose t h a t the s t a t e m e n t holds for all Lie algebras of dimension less t h a n dim L. Let K be a m a x i m a l proper subalgebra of L. We consider the adjoint representation of K on L, a d L " K ~ g[(L). Let x C K , then a d L x ( K ) C K . So K is a s u b m o d u l e of L. We form the quotient module and get a representation a" K ~ g [ ( L / K ) . 1, a(x) is nilpotent for all x C K .

Ijk dkl -- ~lkjdik -- "/~kdjk -- 0, for 1 _< i, j, l _< m. k=l Which is a system of m 3 linear equations for the m 2 variables dij. This system can be solved by a Gaussian elimination; as a consequence we find an algorithm Derivations which computes a basis of Der(A) for any algebra A. R e m a r k . 1), YiYj - - y j y i . Therefore, d C End(A) if and only if d(yiYj) - d(yi)Yj + yid(yj) for 1 < i < 24 Basic constructions j < m. So in this case we find m2(m + 1)/2 equations (instead of m3). 1 Let L be a Lie algebra.

Let V be the subspace of L spanned y l , . . 10). T h e n x - Y'~i c~ixi is an element of NL(V) if and only if there a r e ~ l m for 1 _< l, rn _< t such that [x, yl] -- ~llyl + . . + ~itYt f o r l - - 1 , . . , t . This amounts to the following linear equations in the variables c~i and flzm: ~ljeij 9 j=l /~mk~lm - - 0 Oti- for 1 _< k <_ n and 1 _< 1 _< t. m=l Again by a Gaussian elimination we can solve these equations. However, we are not interested in the values of the film, so we throw the part of the solution that corresponds to these variables away, and we find a basis of N L ( V ) .