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We give a combinatorial description of the ``$D_{2n}$ planar algebra, by generators and relations. We explain how the generator interacts with the Temperley-Lieb braiding. This shows the previously known braiding on the even part extends to a `braiding up to sign on the entire planar algebra. We give a direct proof that our relations are consistent (using this `braiding up to sign), give a complete description of the associated tensor category and principal graph, and show that the planar algebra is positive definite. These facts allow us to identify our combinatorial construction with the standard invariant of the subfactor $D_{2n}$.
In this paper, the structure of the parafermion vertex operator algebra associated to an integrable highest weight module for simple affine Lie superalgebra $osp(1|2n)$ is studied. Particularly, we determine the generators for this algebra.
We compute the second (and the first) cohomology groups of $^*$-algebras associated to the universal quantum unitary groups of not neccesarily Kac type, extending our earlier results for the free unitary group $U_d^+$. The extended setup forces us to
This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (fo
We study the $C^*$-algebra $mathbb{E}_{mathscr{M}}$ of a smooth 4-dimensional manifold $mathscr{M}$ introduced by Gabor Etesi. It is proved that the $mathbb{E}_{mathscr{M}}$ is a stationary AF-algebra. We calculate the topological and smooth invarian
We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${mathbb Z}$-graded algebras. As our main application of this theorem, we obtain isomorph