Abstract:
Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $P(\mathfrak{g})$ be the corresponding Poisson algebra.
With each regular element $a\in \mathfrak{g}$, the argument shift method associates a commutative
subalgebra $F(a)\subset P(\mathfrak{g})$, whose transcendence degree is maximal possible, i.e.,
is equal to the dimension of a Borel subalgebra of $\mathfrak{g}$. When a tends to a singular
element in a proper way, the subalgebra $F(a)$ tends to some commutative subalgebra
of the same transcendence degree. The cases when a tends to a singular element
remaining in the same Cartan subalgebra, were investigated in old works of the
speaker (1990) and V.V. Shuvalov (2002). Some other cases will be discussed in the
talk. An interesting problem is to describe the variety of integrable quadratic
Hamiltonians arising in this way.
