Comparison of Admissibility Conditions for Cyclotomic Birman–wenzl–murakami Algebras

Cyclotomic Birman–Wenzl–Murakami (BMW) algebras are BMW analogues of cyclotomic Hecke algebras [2, 1]. They were defined by Häring–Oldenburg in [7] and have recently been studied by three groups of mathematicians: Goodman and Hauschild–Mosley [4, 5, 6, 3], Rui, Xu, and Si [9, 8], andWilcox and Yu [10, 11, 12, 13]. A peculiar feature of these algebras is that it is necessary to impose “admissibility" conditions on the parameters entering into the definition of the algebras in order to obtain a satisfactory theory. There is no one obvious best set of conditions, and the different groups studying these algebras have proposed different admissibility conditions and have chosen slightly different settings for their work. Under their various admissibility hypotheses on the ground ring, the several groups of mathematicians mentioned above have obtained similar results for the cyclotomic BMW algebras, namely freeness and cellularity. In addition, Goodman & Hauschild– Mosley and Wilcox & Yu have shown that the algebras can be realized as algebras of tangles, while Rui et. al. have obtained additional representation theoretic results, for example, classification of simple modules and semisimplicity criteria. However, it has been difficult to compare the results of the different investigations because of the different settings. The purpose of this note is to show that the admissibility condition proposed by Rui and Xu [9] is equivalent to the condition proposed by Wilcox and Yu [10]. As a result, one has a consensus setting for the study of cyclotomic BMW algebras. Furtherbackground on cyclotomic BMWalgebras,motivation for the study of these algebras, relations to other mathematical topics (quantum groups, knot theory), and further literature citations can be found in [5] and in the other papers cited above.