Mathematics – Quantum Algebra
Scientific paper
2010-02-12
Mathematics
Quantum Algebra
LaTeX 35 pages
Scientific paper
A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties. Equivalently, they are indexed by broken lines $L$. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another. Compatible pairs can be written down. The algebras are equal to their upper cluster algebras. The variables of the quantum seeds are given by elements of the dual canonical basis. In this revised version, the word `quantum' has been added to the title. The manuscript has been restructured. Some proofs have been expanded, some have been rewritten, and some omitted proofs have been furnished. Several typos have been corrected. Some theorems have been expanded and some structures given only in prose have been properly defined.
Jakobsen Hans Plesner
Zhang Hechun
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