Elliptic Ding–Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator
gp(x) := Θp(qx)Θp(t −1x)Θp(q−1tx) Θp(q−1x)Θp(tx)Θp(qt−1x) . Here we have used the notation above and assume |q| < 1, |p| < 1. We define the elliptic Ding-Iohara algebra U(q, t, p) to be the associative C-algebra generated by {xn (p)}n∈Z, {ψ± n (p)}n∈Z and γ subject to the following relation : we set γ as the central, invertible element and currents to be x±(p; z) := ∑ n∈Z x ± n (p)z −n, ψ±(p; z) := ∑ n∈Z ψ ± n (p)z −n. [ψ±(p; z), ψ±(p;w)] = 0, ψ(p; z)ψ−(p;w) = gp(γz/w) gp(γ−1z/w) ψ−(p;w)ψ+(p; z), ψ±(p; z)x(p;w) = gp ( γ± 1 2 z w ) x+(p;w)ψ±(p; z), ψ±(p; z)x−(p;w) = gp ( γ∓ 1 2 z w )−1 x−(p;w)ψ±(p; z), x±(p; z)x±(p;w) = gp ( z w )±1 x±(p;w)x±(p; z), [x(p; z), x−(p;w)]