A classical result in Lie theory stipulates that a simple (but not simply laced) finite dimensional Lie algebra can be constructed as the subalgebra of a Lie algebra of type ADE fixed by an admissible automorphism of the Dynkin diagram of the latter. This construction is called a "folding" (since the Dynkin diagram of the fixed point subalgebra is obtained by "folding" the Dynking diagram of type ADE along the orbits of the automorphim) and extends to Kac-Moody Lie algebras. Although this folding construction does not admit direct quantum analogues, it can be shown that there exists an embedding of crystals for the corresponding Langlands dual Lie algebras. In this talk we will introduce algebraic analogues and generalizations of foldings to the quantum setting which yield new flat quantum deformations of non-semisimple Lie
algebras and of Poisson algebras (joint work with A. Berenstein).

We study a 1+1-dimensional directed polymer in a random
environment on the integer lattice with log-gamma distributed
weights and both endpoints of the polymer path fixed.
We show that under appropriate boundary conditions
the fluctuation exponents for the free energy and
the polymer path take the values conjectured in the
theoretical physics literature. Without the boundary
we get the conjectured upped bounds on the exponents.