Lie algebras and cohomology of congruence subgroups for SLn (R)
- Autores: Jonathan Lopez
- Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 218, Nº 2, 2014, págs. 256-268
- Idioma: inglés
- DOI: 10.1016/j.jpaa.2013.05.011
- Enlaces
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Resumen
Let R be a commutative ring that is free of rank k as an abelian group, p a prime, and SLn(R) the special linear group. We show that the Lie algebra associated to the filtration of SLn(R) by p-congruence subgroups is isomorphic to the tensor product sln(R.Z Z/p).Fp tFp[t], the Lie algebra of polynomials with zero constant term and coefficients n �~ n traceless matrices with entries polynomials in k variables over Fp.
We also use the underlying group structure to obtain several homological results. For example, we compute the first homology group of the level p-congruence subgroup for n . 3. We show that the cohomology groups of the level pr-congruence subgroup are not finitely generated for n = 2 and R = Z[t]. Finally, we show that for n = 2 and R = Z[i] (the Gaussian integers) the second cohomology group of the level pr-congruence subgroup has dimension at least two as an Fp-vector space.
