We indicate which symmetrizable locally compact affine or hyperbolic Kac-Moody groups satisfy Kazhdan’s Property (T), and those that satisfy its strong negation, the Haagerup property. This reveals a new class of hyperbolic Kac-Moody groups satisfying the Haagerup property, namely symmetrizable locally compact Kac-Moody groups of rank 2 or of rank 3 noncompact hyperbolic type. These groups thus...

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...

Mutation of skew-symmetrizable matrices is a fundamental operation that first arose in Fomin-Zelevinsky’s theory of cluster algebras; it also appears naturally in many different areas of mathematics. In this paper, we study mutation classes of skew-symmetrizable 3 × 3 matrices and associated graphs. We determine representatives for these classes using a natural minimality condition, generalizin...

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class ...

We introduce the notions of symmetric and symmetrizable representations $\text{SL}_2(\mathbb{Z})$. The linear $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are have congruence kernel. Conversely, one may also reconstruct data finite-dimensional symmetric, By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry some Weil at prime power levels, we prove that all symmetrizable. pro...

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how...