## FEDERICO MARIA STARA

PhD Graduate**PhD program:**: XXXIII

**supervisor**: A. De Sole

**advisor**: P. Papi

**Thesis title:**On Panyushev's Rootlets for Infinitesimal Symmetric Spaces

Let g be a simple Lie algebra, b a fixed Borel subalgebra, R the corresponding root system and W the associated Weyl group.
In the first part of this work we decompose the poset of abelian ideals of b into peculiar subposets,
which turn out to be isomorphic to right coset representatives obtained from specific subgroups of W. We use this to give a new proof
of the Panyushev's one-to-one correspondence between maximal abelian ideals of b and long simple roots of R.
In the second part of the work we extend the study to the case where g = g_0 + g_1 is a Z_2-graded Lie algebra.
We study the poset of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0,
and we decompose it, in the semisimple cases, into specific subposets, that we show are isomorphic to right coset
representatives obtained from specific subgroups of the associated affine Weyl group.