On Z-graded loop Lie algebras, loop groups, and Toda equations

Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with integrable Z-gradations is discussed.Comment: 24 pages, talk given at the Workshop "Classical and Quantum Integrable Systems" (Dubna, January, 2007

On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras

We define the twisted loop Lie algebra of a finite dimensional Lie algebra $\mathfrak g$ as the Fr\'echet space of all twisted periodic smooth mappings from $\mathbb R$ to $\mathfrak g$. Here the Lie algebra operation is continuous. We call such Lie algebras Fr\'echet Lie algebras. We introduce the notion of an integrable $\mathbb Z$-gradation of a Fr\'echet Lie algebra, and find all inequivalent integrable $\mathbb Z$-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.Comment: 26 page

Enumeration of quarter-turn symmetric alternating-sign matrices of odd order

It was shown by Kuperberg that the partition function of the square-ice model related to the quarter-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with the quarter-turn symmetric alternating-sign matrices of odd order, and show that the partition function of this model can be also written in a similar way. This allows to prove, in particular, the conjectures by Robbins related to the enumeration of the quarter-turn symmetric alternating-sign matrices.Comment: 11 pages, 13 figures; minor correction