We construct the family of spin chain Hamiltonians, which have affine quantum group symmetry U q ˆ g. Their eigenvalues coincide with the eigen-values of the usual spin chain Hamiltonians, but have the degeneracy of levels, corresponding to affine U q ˆ g. The space of states of these spin chains is formed by the tensor product of fully reducible representations of quantum group. The fermionic ...

We provide a new presentation for the annular braid group. The annular braid group is known to be isomorphic to the finite type Artin group with Coxeter graph Bn. Using our presentation, we show that the annular braid group is a semidirect product of an infinite cyclic group and the affine Artin group with Coxeter graph Ãn−1. This provides a new example of an infinite type Artin group which inj...

‎Highest weight modules of the double affine Lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎Singular vectors of Verma modules are‎ ‎determined using a similar condition with horizontal affine Lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

Affine Deligne-Lusztig varieties are analogs of DeligneLusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper [5] by Haines, Kottwitz, Reuman, and the first named author, about the question which affine DeligneLusztig varieties (for a split group and a basic σ-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a...

Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In t...

One of the most appealing results metric-affine gauge theory gravity is a close parallel between Riemann curvature two-form and Cartan torsion two-form: While former field strength Lorentz-group connection one-form, latter can be understood as coframe one-form. This parallel, unfortunately, not fully established until one adopts Trautman's idea introducing an affine-vector-valued zero-from, mea...

In this short note we provide an example of a semi-linear group G which does not admit a semi-linear affine embedding; in other words, there is no semi-linear isomorphism between topological groups f : G → G′ ⊆ Mm, such that the group topology on G′ coincides with the subspace topology induced by Mm. Let M be an o-minimal structure. By “definable” we mean “definable in M” possibly with paramete...

It is known that Seifert-van Kampen theorem (for “good” topological spaces) can be showed by arguing the category of covering spaces. Similar arguments should be valid for abstract Galois categories (which Mochizuki calls ”connected anabelioids”) and neutral Tannakian categories. But when we try to state the theorem, the problem is the existence of amalgams (in other words, fibered coproducts) ...