arXiv subjects
Publications and source records attributed to Stephen T. Moore.
We give an alternative presentation of the small version of the reflection equation algebra associated to $GL_N$ at both odd and even roots of unity, and use our presentation to classify its irreducible representations. We then describe a family of algebras generalizing the small reflection equation algebra, and consider their application to the study of module categories over $U_q(sl_N)$ fusion categories.
We continue our study of Hilbert space representations of the Reflection Equation Algebra, again focusing on the algebra constructed from the $R$-matrix associated to the $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We develop a form of highest weight theory and use it to classify the irreducible bounded $*$-representations of the reflection equation algebra.
We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We consider the Poisson structure appearing as the classical limit of the $R$-matrix, and parametrize the symplectic leaves explicitly in terms of a type of matrix we call a shape matrix. We then introduce a quantized version of the shape matrix for the REA, and show that each irreducible representation of the REA has a unique shape.
We prove a version of Sylvester's law of inertia for the Reflection Equation Algebra (=REA). We will only be concerned with the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$. For $q$ positive, this particular REA comes equipped with a natural $*$-structure, by which it can be viewed as a $q$-deformation of the $*$-algebra of polynomial functions on the space of self-adjoint $N$-by-$N$-matrices. We will show that this REA satisfies a type $I$-condition, so that its irreducible representations can in principle be classified. Moreover, we will show that, up to the adjoint action of quantum $GL(N,\mathbb{C})$, any irreducible representation of the REA is determined by its \emph{extended signature}, which is a classical signature vector extended by a parameter in $\mathbb{R}/\mathbb{Z}$. It is this latter result that we see as a quantized version of Sylvester's law of inertia.
We review the classification of positive extremal traces on the generic infinite Temperley-Lieb algebra, and then extend the classification to the non-semisimple root of unity case. As a result, we obtain Hilbert space structures on the full infinite Temperley-Lieb algebra at roots of unity.
We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for $TL_{n}$ that generalizes to give extensions of $TL_{\infty}$ representations. Finally we define a generalization of the spin chain representation and conjecture a generalization of Schur-Weyl duality.
We give diagrammatic formulae for morphisms between indecomposable representations of $\bar{U}_{q}(\mathfrak{sl}_{2})$ appearing in the decomposition of $\mathbb{C}^{\otimes 2n}$, including projections and second endomorphisms on projective indecomposable representations.