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Universal geometric coefficients for the oncepunctured torus
, 2012
"... Abstract. We construct universal geometric coefficients, over Z, Q, and R, for cluster algebras arising from the oncepunctured torus. We verify that the oncepunctured torus has a property called the Null Tangle Property. The universal geometric coefficients over Z and Q are then given by the shear ..."
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Abstract. We construct universal geometric coefficients, over Z, Q, and R, for cluster algebras arising from the oncepunctured torus. We verify that the oncepunctured torus has a property called the Null Tangle Property. The universal geometric coefficients over Z and Q are then given by the shear coordinates of certain “allowable ” curves in the torus. The universal geometric coefficients over R are given by the shear coordinates of allowable curves together with the normalized shear coordinates of certain other curves each of which is dense in the torus. We also construct the mutation fan for the oncepunctured torus and recover a result of Nájera on gvectors. Contents
Universal geometric cluster algebras
, 2012
"... We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually Z, Q, or R. We broaden the definition of geometric cluster algebras slightly over the us ..."
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Cited by 4 (3 self)
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We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually Z, Q, or R. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal geometric coefficients is equivalent to finding an Rbasis for B (a “mutationlinear” analog of the usual linearalgebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan FB, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between FB and gvectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.