Classically, the Birman-Wenzl-Murakami (BMW) algebra \(W_{n,S}\) [J. S. Birman and H. Wenzl, Trans. Am. Math. Soc. 313, No.1, 249–273 (1989; Zbl 0684.57004); J. Murakami, Osaka J. Math. 24, 745–758 (1987; Zbl 0666.57006)] is defined as a quotient of \(SB_ n\) by generators and relations, where \(S\) is a commutative unital ring with units \(q\),\(\rho\) and \(\delta_0\) satisfying \(\rho^{-1}-\rho=(q^{-1}-q)(\delta_0-1)\). It is closely related to the Kauffman polynomial [L. H. Kauffman, Trans. Am. Math. Soc. 318, No.2, 417–471 (1990; Zbl 0763.57004)], which can be obtained via a suitable Markov trace on the BMW algebra. On the other hand, H. R. Morton and P. Traczyk [in ‘Knots and algebras’, eds. E. Martin-Peindador, A. Rodez Usan, University of Zaragoza, 1990, 201–220 (1990)] defined a certain algebra of tangles, known as the Kauffman tangle algebra, \(KT_{n,S}\), as an \(S\)-algebra of framed \((n,n)\)-tangles modulo Kauffman skein relations (crossing relation, untwisting relation, free loop relation). H. Morton and A. Wassermann [A basis for the Birman-Wenzl algebra, unpublished, (1999)] proved that \(KT_{n,S}\) is a free \(S\)-module of rank \((2n-1)!!\) isomorphic to \(W_{n,S}\).
F. M. Goodman and H. Hauschild [Fundam. Math. 190, 77–137 (2006; Zbl 1100.57008)] proved the analogue of this result for affine BMW algebras. On the geometric side, the affine Kauffman tangle algebra, \(\widehat{KT}_{n,S}\), is an algebra of \((n,n)\) tangles in \(A\times{}I\) (\(A\) is the annulus), or equivalently tangles in a space with a fixed “flagpole”, while algebraically, the affine BMW algebra \(\widehat{W}_{n,S}\) has additional generators and relations. The appropriate skein relations are altered, and there are additional parameters \(\delta_{j}\) corresponding to free loops winding \(j\) times around the flagpole.
The current paper, along with its second part [Algebr. Representation Theory, to appear], carries out the same program for the cyclotomic analogues. The cyclotomic BMW algebra \(W_{n,S,r}\) and cyclotomic Kauffman tangle algebra \(KT_{n,S,r}\) are dependent on invertible \(\rho\), \(q\), \(u_1,\ldots,u_{r}\) and parameters \(\delta_{j}\) (\(j\geq 0\)). They are defined as quotients of the affine versions by an additional “cyclotomic” relation, which for the BMW algebra has the form \((y_1-u_1)(y_1-u_2)\cdots(y_1-u_{r})=0\) while for the Kauffman tangle algebra, it is an analogous skein relation.
The main theorem states that under a suitable admissibility condition on \(S\) and the parameters (the form of which is due to Wilcox and Yu [Commun. Algebra, to appear]), \(W_{n,S,r}\) is a free \(S\)-module of rank \(r^n(2n-1)!!\) isomorphic to \(KT_{n,S,r}\). Applying techniques of affine and cyclotomic Hecke algebras to the case of cyclotomic BMW algebras, the authors also find an explicit basis and show the existence of the appropriate Markov trace. This is related to the \(R\)-matrix representation of the type B Artin braid groups.