Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Alternative formulation for the operator algebra over the space of paths
  in a ADE $SU(3)$ graph   In this work we discuss the elements required for the construction of the
operator algebra for the space of paths over a simply laced $SU(3)$ graph.
These operators are an important step in the construction of the bialgebra
required to find the partition functions of some modular invariant CFTs. We
define the cup and cap operators associated with back-and-forth sequences and
add them to the creation and annihilation operators in the operator algebra as
they are required for the calculation of the full space of essential paths
prescribed by the fusion algebra. These operators require collapsed triangular
cells that had not been found in previous works; here we provide explicit
values for these cells and show their importance in order for the cell system
to fulfill the Kuperberg relations for $SU(3)$ tangles. We also find that
demanding that our operators satisfy the Temperley-Lieb algebra leads one
naturally to consider operators that create and annihilate closed triangular
sequences, which in turn provides an alternative the cup and cap operators as
they allow one to replace back-and-forth sequences with closed triangular ones.
We finally show that the essential paths obtained by using closed triangles are
equivalent to those obtained originally using back-and-forth sequences.

1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc