2011/11/01 by Reeves, Andrew
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1111.0145
The symplectic blob algebras are a family of finite dimensional noncommutative algebras over ℤ[X1,X2,X3,X4,X5,X6] 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 A ⊗ \sba for each n ∈ ℕ, for A a particular commutative ring with indeterminates. The form of this representation is motivated by the XXZ representation of the Temperley-Lieb algebra \TLn \citejimbo86 and the related Martin-Woodcock representation of the blob algebra bn \citemartinwoodcock2003. For k an algebraically closed field, and for any (δ,δL,δR,κL,κR,κ) ∈ k6, the algebra \sba specialises to a k-algebra \sba(δ,δL,δR,κL,κR,κ). For any such specialisation, our representation passes to a \sba(δ,δL,δR,κL,κR,κ)-module V(n).