We introduce quasi-homomorphisms of cluster algebras, a flexible notion of a map between cluster algebras of the same type (but with different coefficients). The definition is given in terms of seed orbits, the smallest equivalence classes of seeds on which the mutation rules for non-normalized seeds are unambiguous. We present examples of quasi-homomorphisms involving familiar cluster algebras, such as cluster structures on Grassmannians, and those associated with marked surfaces with boundary. We explore the related notion of a quasi-automorphism, and compare the resulting group with other groups of symmetries of cluster structures. For cluster algebras from surfaces, we determine the subgroup of quasi-automorphisms inside the tagged mapping class group of the surface.
Bibliographical noteFunding Information:
I thank Greg Muller and Gregg Musiker for helpful conversations. I especially thank Sergey Fomin for his guidance and encouragement. This work was supported by a graduate fellowship from the National Physical Science Consortium and National Science Foundation grant DMS-1361789 . While in the midst of carrying out this work, I learned that Thomas Lam and David Speyer had independently obtained results similar to Corollary 4.5 and Remark 4.6 .
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- Cluster algebra
- Cluster modular group
- Seed orbit
- Tagged mapping class group