Quantum Algebra and Quantum Topology
نویسنده
چکیده
I work in an area at the intersection of representation theory, low-dimensional topology, higher category theory, and operator algebras which is often referred to as quantum algebra or quantum topology. A practical description of this field is that it consists of the mathematics which is descended from the Jones polynomial [Jon85]. The unifying idea behind quantum topology is to consider a functor from a tensor category (or more generally an n-category) of a topological nature to a tensor category of an algebraic nature [Ati88, Wit89, RT90, RT91]. This allows one to construct topological invariants from algebraic structures, and to use topological arguments to prove theorems in algebra. Typically we write algebraic expressions in a 1-dimensional way. In ordinary algebra, you can multiply on the left or on the right, but you can’t multiply above or below. The key observation of quantum algebra is that certain algebraic structures become more clear if you write them in the plane, or more generally on higher dimensional manifolds. A striking example from my own work is that the proof of Radford’s theorem on the fourth power of the antipode in a Hopf algebra [Rad76] (and more generally, the quadruple dual of an object in a finite tensor category [ENO04]) is best interpreted as taking place on the following 3-framed disc [DSPS].
منابع مشابه
Hermitian metric on quantum spheres
The paper deal with non-commutative geometry. The notion of quantumspheres was introduced by podles. Here we define the quantum hermitianmetric on the quantum spaces and find it for the quantum spheres.
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