A finite Hecke algebra $H_n(q)$ of type $A$ is an associated algebra generated by $T_1, \ldots, T_{n-1}$ with relations \begin{align} & T_i T_j = T_j T_i, |i-j|>1, \\ & T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1}, \\ & (T_i-q)(T_i+1) = 0. \end{align}
An affine Hecke algebra $\widehat{H_n}(q)$ of type $A$ is an associated algebra generated by $T_1, \ldots, T_{n-1}, X_1^{\pm 1}, \ldots, X_n^{\pm 1}$ with relations \begin{align} & T_i T_j = T_j T_i, |i-j|>1, \\ & T_i T_{i+1} T_i = T_{i+1} T_i T_{i+1}, \\ & (T_i-q)(T_i+1) = 0, \\ & X_i^{\pm 1} X_k^{\pm 1} = X_k^{\pm 1} X_i^{\pm 1}, 1 \leq i, k \leq n, \\ & X_k X_k^{-1} = X_k^{-1} X_k =1, \\ & T_i X_k = X_k T_i, |i-k|>1, \\ & X_{i+1} = q^{-1} T_i X_i T_i, 1 \leq i < n. \end{align} Is $H_n(q)$ a subalgebra of $\widehat{H_n}(q)$? Thank you very much.