Related papers: A note on generators for finite depth subfactor pl…

The Kuperberg Program asks to find presentations of planar algebras and use these presentations to prove results about their corresponding categories purely diagrammatically. This program has been completed for index less than 4 and is…

Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\GG(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We…

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will…

Given a finite index subfactor, we show that the {\em affine morphisms at zero level} in the affine category over the planar algebra associated to the subfactor is isomorphic to the fusion algebra of the subfactor as a *-algebra. This…

We present more planar algebraic construction of subfactors than those of Guionet-Jones-Shlyakhtenko-Walker and Kodiyalam-Sunder which start from a subfactor planar algebra and give in a direct way a subfactor of the same standard invariant…

We consider inclusions of type $(P\otimes A)^G\subset(P\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\subset B$. In the…

We give generators and relations for the planar algebras corresponding to $ADE$ subfactors. We also give a basis and an algorithm to express an arbitrary diagram as a linear combination of these basis diagrams.

Using an analogue of the Guionnet-Jones-Shlaykhtenko construction for graphs we show that their construction applied to any subfactor planar algebra of finite depth yields an inclusion of interpolated free group factors with finite…

We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the `extended Haagerup' principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range…

This paper introduces the cyclic subfactors, generalizing the cyclic groups as the subfactors generalize the groups, and generalizing the natural numbers as the maximal subfactors generalize the prime numbers. On one hand, a theorem of O.…

Let $F$ be a field and let $E$ be an \'etale algebra over $F$, that is, a finite product of finite separable field extensions $E = F_1 \times \dots \times F_r$. The classical primitive element theorem asserts that if $r = 1$, then $E$ is…

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of…

Let $S$ be a submonoid of a free Abelian group of finite rank. We show that if $k$ is a field of prime characteristic such that the monoid $k$-algebra $k[S]$ is split $F$-regular, then $k[S]$ is a finitely generated $k$-algebra, or…

Let $k$ be a number field, suppose that $B$ is a central simple division algebra over $k$, and choose any maximal order $\mathcal{D}$ of $B$. The object of this paper is to show that the group $\mathcal{D}_S^*$ of $S$-units of $B$ is…

Generalizing Krieger's finite generation theorem, we give conditions for an ergodic system to be generated by a pair of partitions, each required to be measurable with respect to a given sub-algebra, and also required to have a fixed size.

We canonically associate to any planar algebra two type II_{\infty} factors M_{+} and M_{-}. The subfactors constructed previously by the authors in a previous paper are isomorphic to compressions of M_{+} and M_{-} to finite projections.…

We prove the following. Let $R$ be a Noetherian ring, $B$ a finitely generated $R$-algebra, and $A$ a pure $R$-subalgebra of $B$. Then $A$ is finitely generated over $R$.

We prove that the infinite-dimensional Lie algebra of polynomial vector fields on the affine space $\KK^n$ is generated by two explicitly given elements.

Let $K/k$ be a finite extension of a global field. Such an extension can be generated over $k$ by a single element. The aim of this article is to prove the existence of a "small" generator in the function field case. This answers the…

It is shown that any finite dimensional simple Lie superalgebra over an algebraically closed field of characteristic 0 is generated by 2 elements.