Journal of Algebraic Combinatorics

, Volume 29, Issue 3, pp 315-335

First online:

The crystal commutor and Drinfeld’s unitarized R-matrix

  • Joel KamnitzerAffiliated withAmerican Institute of Mathematics Email author 
  • , Peter TingleyAffiliated withDepartment of Mathematics, UC Berkeley

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Drinfeld defined a unitarized R-matrix for any quantum group \(U_{q}(\mathfrak {g})\) . This gives a commutor for the category of \(U_{q}(\mathfrak {g})\) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives \(U_{q}(\mathfrak {g})\) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer’s construction agrees with Drinfeld’s commutor. We then describe the action of Drinfeld’s commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor.


Coboundary category Quantum group R-matrix Crystal basis