Cluster automorphisms and quasi-automorphisms

Abstract

We study the relation between the cluster automorphisms and the quasiautomorphisms of a cluster algebra A. We proof that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of A is isomorphic to a subgroup of the cluster automorphism group of Atriv, and the two groups are isomorphic if A has principal or universal coefficients; here Atriv is the cluster algebra with trivial coefficients obtained from A by setting all frozen variables equal to the integer 1.We also compute the quasi-automorphism group of all finite type and all skewsymmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of Atriv.

Quasi-homomorphisms of quantum cluster algebras

“…Therefore we have an isomorphism Aut(A q ) ∼ = Aut(A). On the other hand, it is proved in [CS19] that QAut(A) ∼ = Aut(A). By using the fact that the Lambda matrix is arising from the triangulation, the method used in [CS19] can be extended to the quantum case, and we have an isomorphism of groups: QAut(A q ) ∼ = Aut(A q ).…”

“…On the other hand, it is proved in [CS19] that QAut(A) ∼ = Aut(A). By using the fact that the Lambda matrix is arising from the triangulation, the method used in [CS19] can be extended to the quantum case, and we have an isomorphism of groups: QAut(A q ) ∼ = Aut(A q ). To sum up, for (almost all) the surface cluster algebras, both for the quantum case and the non-quantum case, the groups we mentioned above are all isomorphic.…”

“…The concept was generalized to quasi-automorphisms of cluster algebras by Fraser in [F16]. More relations between cluster automorphisms and quasi-automorphisms were discovered in [CS19]. Both the automorphisms and quasi-automorphisms of cluster algebras are essential tools in the study of symmetries of cluster algebras.…”

Twist Automorphisms and Poisson Structures

“…When one identifies F A (t) and F A (t ′ ) using the mutation map µ * , a twist endomorphism tw A on U A becomes the same as a quasi-homomorphism for a normalized seed pattern in the sense of [14]. [6] studied the group of the twist automorphisms on U A for principal coefficient cases and several other special cases.…”

“…The set of theta functions [25]: the set of the theta functions, under the assumption that they are Laurent polynomials, 6 satisfies the above assumption. They are known to form a basis of U A (t) under appropriate conditions, for example, when t is injective-reachable.…”