A Tensor Space Representation of the Symplectic Blob Algebra

Abstract

The symplectic blob algebras are a family of finite dimensional noncommutative algebras over $\mathbb{Z}[X_1,X_2,X_3,X_4,X_5,X_6]$ that can be defined in terms of planar diagrams in a way that extends the Temperley-Lieb and (ordinary) blob algebras. In this paper we construct a new "tensor space" representation of the symplectic blob algebra $\mathcal{A} \otimes \sba$ for each $n\in \mathbb{N}$, for $\mathcal{A}$ a particular commutative ring with indeterminates. The form of this representation is motivated by the XXZ representation of the Temperley-Lieb algebra $\TL_n$ \cite{jimbo86} and the related Martin-Woodcock representation of the blob algebra $b_n$ \cite{martinwoodcock2003}. For $k$ an algebraically closed field, and for any $(\delta ,{\delta }_L,{\delta }_R,{\kappa }_L,{\kappa }_R,\kappa )\in k^6$, the algebra $\sba$ specialises to a $k$-algebra $\sba(δ,δ_L,δ_R,κ_L,κ_R,κ)$. For any such specialisation, our representation passes to a $\sba(δ,δ_L,δ_R,κ_L,κ_R,κ)$-module $\mathcal{V}(n)$.

19 pages, 4 figures

References in corpus (4)

• The two-boundary Temperley-Lieb algebra
• The Temperley-Lieb algebra and its generalizations in the Potts and XXZ models
• On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary
• On the non-generic representation theory of the symplectic blob algebra

Cited by in corpus (2)

• Tilting Modules for the Symplectic Blob Algebra
• How to fold a spin chain: Integrable boundaries of the Heisenberg XXX and Inozemtsev hyperbolic models