What does this research mean for the field?

For a multiplicity space of dimension 2 or greater, the category of internal polynomial right modules of the Berenstein-Zwicknagl quantum symmetric algebra on multiple standard copies is not equivalent to the corresponding classical module category. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine the component structure of quantum symmetric algebras on multiple standard copies and test whether their module categories match classical counterparts.

September 8, 2026Open Access

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

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To determine the component structure of quantum symmetric algebras on multiple standard copies and test whether their module categories match classical counterparts.
Analyzed the Berenstein-Zwicknagl quantum symmetric algebra $S_\sigma(E \otimes U)$ over the generic polynomial Hecke category over $\mathbb{C}(q)$ with multiplicity space $U$.
Constructed an explicit two-dimensional Hecke braid defect to evaluate morphism spaces between projective covers.
Identified that only one-row and one-column representations survive in degree $n \ge 2$, with respective multiplicities $\mathrm{Sym}^n U$ and $\wedge^n U$.
Characterized the algebra as a fiber product of two diagonal algebras over its square-zero degree-one truncation.
Showed that for $\dim U \ge 2$, the internal polynomial right module category is not equivalent to the classical module category due to a vanishing Hom space obstruction.

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Implication

Theoretical analysis demonstrates categorical non-equivalence in quantum symmetric algebras over multiple copies, highlighting a structural divergence from classical module categories.

Cite This Study

Alper Ferudun (2026) studied this question.

synapsesocial.com/papers/6a9fd83b58e84d0ff5b47a0a https://doi.org/10.5281/zenodo.22635787

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Implication

Theoretical analysis demonstrates categorical non-equivalence in quantum symmetric algebras over multiple copies, highlighting a structural divergence from classical module categories.

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

Key Points

Discussion

Implication

Cite This Study

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

Key Points

Discussion

Implication

Cite This Study

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