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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 contained in 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.
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The Morita equivalences of classical Brauer algebras and classical Birman-Murakami-Wenzl algebras have been well studied. Here we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and Birman-Murakami-Wenzl algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of some Coxeter groups, respectively.
In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of g and give explicit details on when it may be maximal in g.
Hecke-Hopf algebras were defined by A. Berenstein and D. Kazhdan. We give an explicit presentation of an Hecke-Hopf algebra when the parameter $m_{ij},$ associated to any two distinct vertices $i$ and $j$ in the presentation of a Coxeter group, equals $4,$ $5$ or $6$. As an application, we give a proof of a conjecture of Berenstein and Kazhdan when the Coxeter group is crystallographic and non-simply-laced. As another application, we show that another conjecture of Berenstein and Kazhdan holds when $m_{ij},$ associated to any two distinct vertices $i$ and $j,$ equals $4$ and that the conjecture does not hold when some $m_{ij}$ equals $6$ by giving a counterexample to it.
Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.
We give a full description of locally finite p-groups G such that the normalized group of units V(FG) of the group algebra FG over a field F of characteristic p has exponent 4.
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