Related papers: On a presentation of the spin planar algebra
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…
Motivated by Jones' braid group representations constructed from spin models, we define {\sl a Jones pair} to be a pair of $\nbyn$ matrices $(A,B)$ such that the endomorphisms $X_A$ and $\D_B$ form a representation of a braid group. When…
For the simple Lie algebra $g = sl(n,C)$ we we find a set of generators and relations for the classical family algebra $(End(g)\otimes S(g))^G$ as an algebra over the ring $I(g)$. From these we can then determine a $I(g)$-linear basis of…
Circuit algebras, introduced by Bar-Natan and the first author, are a generalization of Jones's planar algebras, in which one drops the planarity condition on "connection diagrams". They provide a useful language for the study of virtual…
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
We develop a random model for relation algebras. We prove some preliminary results and pose questions that lay out a new direction of research.
In these notes we generalize the theory of graphical functions from scalar theories to theories with spin.
In the previous article we introduced the new concept of mixed representations of quivers and described the generators of their algebras of invariants. In this article we describe the defining relations of these algebras. Some applications…
Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in…
In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In…
We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar…
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.
When utilizing a cluster decomposible relativistic scattering formalism, it is most convenient that the covariant field equations take on a linear form with respect to the energy and momentum dispersion on the fields in the manner given by…
A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
The Algebra of Physical Space (APS) is used to explore the Constructive Standard Model (CSM) of particle physics. Namely, this paper connects the spinor formalism of the APS to massive amplitudes in the CSM. A novel equivalency between…
An intriguing correspondence between certain finite planar tessellations and the Descartes circle arrangements is presented. This correspondence may be viewed as a visualization of the spinor structure underlying Descartes circles.
Anchored planar algebras, a generalized notion of Vaughan Jones' planar algebras, have recently seen use in higher category theory, functional analysis, and TQFT applications. These algebras are equipped with a natural 3-dimensional…
We develop an algebraic language theory based on the notion of an Eilenberg--Moore algebra. In comparison to previous such frameworks the main contribution is the support for algebras with infinitely many sorts and the connection to logic…
We introduce a geometric construction which relates to the pentagram map much in the way that a logarithmic spiral relates to a circle. After introducing the construction, we establish some basic geometric facts about it, and speculate on…