Torsion table for the lie algebra ▫$\mathfrak{nil}_n$▫
Lampret, Leon ; Vavpetič, Aleš
V članku študiramo Liejev kolobar ▫$\mathfrak{nil}_n$▫ vseh strogo zgornje trikotnih ▫$n\times n$▫ matrik nad ▫$\mathbb{Z}$▫. Za ▫$n\leq 8$▫ smo izračunali celotno homologijo tega kolobarja. Dokazali ...smo, da v ▫$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$▫ nastopa ▫$p^m$▫-torzija za vse praštevilske potence ▫$p^m\leq n-2$▫. Pri ▫$m=1$▫ je Dwyer pokazal, da je ta meja natančna, tj. ▫$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$▫ ne vsebuje ▫$p$▫-torzije za vse ▫$p>n-2$▫. V splošnem pa za ▫$m>1$▫ meja ni natančna, saj smo pokazali, da ▫$H_\ast(\mathfrak{nil}_8;\mathbb{Z})$▫ vsebuje ▫$8$▫-torzijo. Spotoma smo izpeljali še znano dejstvo, da so rangi prostih delov grup ▫$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$▫ enaki Mahonovim številom (=število permutacij množice ▫$[n]$▫ s ▫$k$▫ inverzijami), preko drugačne izpeljave kot jo je sprva imel Kostant. Na koncu smo določili še algebraično strukturo (kupasti produkti) za ▫$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$▫.
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