It is known that the recently discovered representations of the Artin groups of type A n , the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D n and E n which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I 1 and I 2 with I 2 ⊂ I 1 such that the quotient with respect to I 1 is the Hecke algebra and I 1 /I 2 is a module for the corresponding Artin group generalizing the Lawrence-Krammer representation. Finally we give conjectures on the structure, the dimension and parabolic subalgebras of the BMW algebra, as well as on a generalization of deformations to Brauer algebras for simply laced spherical type other than A n .

Abstract. A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman-Murakami-Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A n−1 . The proof involves a diagrammatic version of the Brauer algebra of type Dn of which the generalized Temperley-Lieb algebra of type Dn is a subalgebra.

The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2 n + 1)n!! − (2 n−1 + 1)n! over a specified commutative ring R, where n!! = 1 • 3 • • • (2n − 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring Z[δ ±1 ]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type.

This study shows that using authentic contexts for learning differential equations in a differentiation-by-interest setting can enhance students' beliefs about the relevance of mathematics. The students in this study were studying advanced mathematics (wiskunde D) at upper secondary school in the Netherlands. These students are often not aware of the relevance of the mathematics they have to learn in school. More insights into the application of mathematics in other sciences can be beneficial for these students in terms of preparation for their future study and career. A course differentiating by student interest with new context-rich curriculum materials was developed in order to enhance students' beliefs about the relevance of mathematics. The intervention aimed at teaching differential equations through guided small-group tasks in scientific, medical or economical contexts. The results show that students' beliefs about the relevance of mathematics improved, and they appreciated experiencing how the mathematics was applied in real-life situations.