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Nil-Coxeter Algebra
In mathematics, the nil-Coxeter algebra, introduced by , is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent. Definition The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by ''u''1, ''u''2, ''u''3, ... with the relations : \begin u_i^2 & = 0, \\ u_i u_j & = u_j u_i & & \text , i-j, > 1, \\ u_i u_j u_i & = u_j u_i u_j & & \text , i-j, =1. \end These are just the relations for the infinite braid group, together with the relations ''u'' = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ..., by adding the relations ''u'' = 0 to the relations of the corresponding generalized braid group. Refere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |