Quantum Symmetric Algebras on Multiple Standard Copies: Structure and Categorical Non-Equivalence

Let \(E\) be the standard object in the generic polynomial Hecke category over \(\mathbb C(q)\) , and let \(U\) be an ordinary finite-dimensional multiplicity space. We determine every homogeneous component of the Berenstein–Zwicknagl quantum symmetric algebra \(A=S_\sigma(E\otimes U)\) : in degree \(n\geq2\) , only the one-row and one-column representations survive, with multiplicities \(\Sym^n U\) and \(\bigwedge^n U\) . We identify \(A\) as a fiber product of two diagonal algebras over its square-zero degree-one truncation. For \(\dim U\geq2\) , its full category of internal polynomial right modules is not equivalent to the corresponding classical module category, even as an abstract abelian category. The obstruction is intrinsic: the directed extension graph identifies two simple objects whose projective covers have a zero Hom space quantumly and a nonzero Hom space classically. The calculation uses an explicit two-dimensional Hecke braid defect and does not assume that an equivalence preserves an augmentation, grading, tensor action, or finite-rank evaluation.