Motivated by our attempts to construct an analogue of the Dirac operator in the setting of , we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra . The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.
In this work, we describe the historic links between the study of -dimensional manifolds (specially knot theory) and the study of the topology of complex plane curves with a particular attention to the role of braid groups and Alexander-like invariants (torsions, different instances of Alexander polynomials). We finish with detailed computations in an example.
If is a non-cyclic finite group, non-isomorphic -sets may give rise to isomorphic permutation representations . Equivalently, the map from the Burnside ring to the rational representation ring of has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of -groups.
We determine the finite groups whose poset of conjugacy classes of subgroups has breaking points. This leads to a new characterization of the generalized quaternion -groups. A generalization of this property is also studied.
We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group over a suitable non-Archimedean field we define a map from the Bruhat-Tits building to the Berkovich analytic space associated with . Composing this map with the projection of to its flag varieties, we define a family of compactifications of . This generalizes results by Berkovich in the case of split groups. Moreover,...
A Brunnian link is a set of n linked loops such that every proper sublink is trivial. Simple Brunnian links have a natural algebraic representation. This is used to determine the form, length and number of minimal simple Brunnian links. Braids are used to investigate when two algebraic words represent equivalent simple Brunnian links that differ only in the arrangement of the component loops.