Invariant lattices, the Leech lattice and its even unimodular analogues in the Lie algebras $A_{p-1}$

For any prime $p>2$ a classification (up to similarity) is given of all invariant integral lattices that correspond to an orthogonal decomposition of the Lie algebra $A_{p-1}$. Even unimodular lattices without roots are distinguished. For $p=5$ they contain the Leech lattice. For some of the resulting lattices the automorphism groups are studied, and lower bounds for the minimal length of vectors are obtained.
Figures: 2.
Bibliography: 17 titles.