Centrality of the quantum hyperdeterminant: constructing the quotient algebra of quantum hypermatrices in which $\operatorname{Det}_q$ becomes central

The algebra $\mathscr{A}_q^{[m]}$ of quantum hypermatrices introduced by Jing and Zhang carries a quantum hyperdeterminant $\operatorname{Det}_q^{[m]}$, a noncommutative $q$-analogue of Cayley's first hyperdeterminant that satisfies the deformed invariance law $\operatorname{Det}_q(B\circ_k A)=\det_q(B)\operatorname{Det}_q(A)$ and maps to $\det_q^{\otimes m}$ under the realignment coproduct — but, unlike the quantum determinant in the quantum matrix algebra, it is not central. The authors conjectured that a nontrivial intermediate quotient $\bar{\mathscr{A}}=\mathscr{A}_q^{[m]}/I$, whose ideal $I$ contains relations making $\operatorname{Det}_q^{[m]}$ central, exists, noted that quantum analogues of Cayley's other hyperdeterminants may provide a solution, and observed that for format $n^4$ quantum Weyl algebra relations, although more numerous than their minimal quadratic relations, appear insufficient. The open problem is that conjecture: determine the additional relations and construct the intermediate quotient. The problem remains open.

Background

An $m$-dimensional hypermatrix (an order-$m$ tensor written as an array) of format $n^m=n\times\cdots\times n$ is an array $A=(a_{i_1\cdots i_m})$ with $1\le i_k\le n$; ordinary $n\times n$ matrices are the case of format $n^2$. In his foundational work on determinants, Cayley defined the first hyperdeterminant of an even-dimensional hypermatrix $A=(a_{i_1\cdots i_{2m}})$ of commuting entries by $\operatorname{Det}(A)=\frac{1}{n!}\sum_{\sigma_1,\ldots,\sigma_{2m}\in S_n}(-1)^{\sum_{i}\ell(\sigma_i)}\prod_{i=1}^{n}a_{\sigma_1(i)\cdots\sigma_{2m}(i)},$ where $S_n$ is the symmetric group and $\ell(\sigma)$ is the number of inversions of $\sigma$ (for format $n^2$ this reduces to the ordinary determinant). Its defining property is relative invariance: if an $n\times n$ matrix $B$ acts on the $k$-th index by contraction, $(B\circ_k A)_{i_1\cdots i_m}=\sum_j b_{i_k j}a_{i_1\cdots j\cdots i_m}$, then $\operatorname{Det}(B\circ_k A)=\det(B)\operatorname{Det}(A)$, which makes $\operatorname{Det}$ a relative invariant for the diagonal action of the direct product $GL_n^{\otimes 2m}$; the modern theory of hyperdeterminants, discriminants, and resultants is developed in the book of Gelfand, Kapranov, and Zelevinsky. Cayley also introduced further hyperdeterminants (a second hyperdeterminant and higher analogues), which belong to the classical invariant theory of multidimensional matrices.

In parallel, on the quantum side, the coordinate ring $\mathrm{Mat}_q(n)$ of quantum matrices — the algebra produced by the FRT construction (of Faddeev, Reshetikhin, and Takhtajan) attached to the standard Drinfeld–Jimbo quantum group $GL_q(n)$, a one-parameter deformation of the ring of polynomial functions on $n\times n$ matrices — is generated by $a_{ij}$, $1\le i,j\le n$, with quadratic relations that Manin characterized conceptually: they are exactly the relations forcing the generators to preserve the quantum exterior algebra $\Lambda_n$, the algebra generated by $x_1,\ldots,x_n$ with relations $x_j\wedge x_i=-q\,x_i\wedge x_j$ for $i<j$ and $x_i\wedge x_i=0$. The quantum determinant $\det_q(A)=\sum_{\sigma\in S_n}(-q)^{\ell(\sigma)}a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)}$ is central in $\mathrm{Mat}_q(n)$ (it commutes with every element) and group-like (compatible with the coproduct via $\Delta(\det_q)=\det_q\otimes\det_q$); adjoining its inverse yields the quantum group $GL_q(n)$, and this centrality underlies the quantum theory of minors, the antipode, and quantum invariant theory. Hashimoto and Hayashi developed the earlier 'quantum multilinear algebra' in the same circle of ideas.

Jing and Zhang (2017) extended the picture to hypermatrices. For a parameter $q$ and format $n^m$ they defined the algebra $\mathscr{A}_q^{[m]}$ of quantum hypermatrices: generators $a_{i_1\cdots i_m}$ subject to quadratic relations obtained by requiring that every matrix realignment of $A$ (regrouping the $m$ indices into one distinguished index plus an ordered $(m-1)$-tuple and reading $A$ as an $n\times n^{m-1}$ matrix) act on tensor powers of the degree-one part of the quantum exterior algebra the way quantum matrices act on $\Lambda_n$. In $\mathscr{A}_q^{[m]}$ they defined the quantum hyperdeterminant $\operatorname{Det}_q^{[m]}(A)=\frac{1}{[n]_{q^2}!}\sum_{\sigma_1,\ldots,\sigma_m\in S_n}(-q)^{\sum_{i=1}^m\ell(\sigma_i)}\prod_{i=1}^{n}a_{\sigma_1(i)\cdots\sigma_m(i)},\qquad [n]_{q^2}!=\sum_{\sigma\in S_n}q^{2\ell(\sigma)},$ proved the deformed invariance $\operatorname{Det}_q(B\circ_k A)=\det_q(B)\operatorname{Det}_q(A)$, invariance under permutations of the $m$ indices, and that $\operatorname{Det}_q$ is a quantum volume element (the coefficient of the top exterior power), and proved that the realignment coproduct $\Delta(a_{\alpha\beta})=\sum_{j=1}^n a_{\alpha j}\otimes a_{j\beta}$ (splitting the index string into two blocks) is an algebra homomorphism under which $\operatorname{Det}_q^{[m]}$ maps to a scalar multiple of $\det_q^{\otimes m}$ in $\mathrm{Mat}_q(n)^{\otimes m}$. For even $m$ this is a $q$-analogue of Cayley's first hyperdeterminant; for odd $m$ it has no nonzero commutative counterpart (it vanishes identically at $q=1$ when the entries commute, but not in $\mathscr{A}_q^{[m]}$), a purely quantum phenomenon.

The defect motivating the present problem: unlike $\det_q$ in $\mathrm{Mat}_q(n)$, $\operatorname{Det}_q^{[m]}$ is not central in $\mathscr{A}_q^{[m]}$ once one goes beyond matrices. The defining relations of $\mathscr{A}_q^{[m]}$ are deliberately minimal — just enough for the Diamond Lemma (the standard rewriting tool that yields a Poincaré–Birkhoff–Witt-type basis of ordered monomials from a confluent presentation) to rearrange the products inside $\operatorname{Det}_q$ and the quantum hyper-Pfaffian — so $\mathscr{A}_q^{[m]}$ is, in the authors' words, a very general uplift of $GL_q(n)^{\otimes m}$. For ordinary quantum matrices (format $n^2$) no additional relations are needed, since the quantum determinant is already central there. For hypermatrices of format $n^4$ the authors observe that replacing their quantum-exterior-algebra relations by those of the quantum Weyl algebra (the $q$-deformed algebra of polynomial differential operators generated by coordinates together with $q$-derivatives) imposes more relations, yet still appears insufficient to make $\operatorname{Det}_q$ central — a suspicion stated without proof. The authors conjecture that an intermediate quotient $\bar{\mathscr{A}}=\mathscr{A}_q^{[m]}/I$ — lying between $\mathscr{A}_q^{[m]}$ and its realignment image in $\mathrm{Mat}_q(n)^{\otimes m}$ — exists, where the ideal $I$ contains relations that make $\operatorname{Det}_q^{[m]}$ central and is trivial in the matrix case; they note that quantum analogues of Cayley's other hyperdeterminants may provide a solution, as indicated by the classical case.

Problem Statement

Let $\mathscr{A}_q^{[m]}$ be the Jing–Zhang algebra of quantum hypermatrices of format $n^m$ ($n\ge 2$, $m\ge 3$ beyond the matrix case, $q$ an indeterminate or generic complex parameter), and let $\operatorname{Det}_q^{[m]}$ be its quantum hyperdeterminant as defined above. The problem is to determine the additional relations that make the quantum hyperdeterminant central — the correct quantum analogue of the coordinate ring of hypermatrices, the 'quantum hyper-$GL$' in which the hyperdeterminant plays the central role that $\det_q$ plays in $GL_q(n)$.

Concretely, following the authors' conjecture: construct explicitly a nontrivial two-sided ideal $I$ of $\mathscr{A}_q^{[m]}$ — trivial in the case of ordinary quantum matrices, where $\det_q$ is already central — such that $\operatorname{Det}_q^{[m]}$ is central in the quotient $\bar{\mathscr{A}}=\mathscr{A}_q^{[m]}/I$ and the quotient is genuinely intermediate between $\mathscr{A}_q^{[m]}$ and its realignment image in $\mathrm{Mat}_q(n)^{\otimes m}$, retaining the hypermatrix structure. These requirements are conditions on the answer, not illustrative qualifiers: in the hypermatrix cases the ideal $I$ must be properly contained in the kernel of the realignment coproduct, so that $\bar{\mathscr{A}}$ is strictly intermediate between $\mathscr{A}_q^{[m]}$ and its realignment image, and $\bar{\mathscr{A}}$ must retain a Poincaré–Birkhoff–Witt-type basis of ordered monomials in the generators, a realignment coproduct that is well defined on the quotient, and the $S_m$-symmetry of the indices. Alternatively, prove that no such intermediate quotient exists.

The source does not delimit the class of relations that the conjectured ideal may contain. Two indications given in the source are that the authors' own defining relations were chosen by a minimality principle (the least relations allowing the Diamond-Lemma rearrangements inside $\operatorname{Det}_q$ and the quantum hyper-Pfaffian), and that quantum analogues of Cayley's other hyperdeterminants may supply the needed relations by analogy with the classical case. For orientation: quotienting by the ideal generated by all commutators $[\operatorname{Det}_q^{[m]},a]$ would impose centrality by fiat, and that ideal lies inside the kernel of the realignment map because $\det_q^{\otimes m}$ is central in $\mathrm{Mat}_q(n)^{\otimes m}$; the mathematical content of the conjecture is to find relations under which the hyperdeterminant becomes central while the quotient remains a quantum hypermatrix algebra meeting those requirements — centrality obtained by fiat alone, as in the commutator quotient just described, does not by itself constitute such a construction.

Two further points are discussed in the source without being posed as separate requirements. For format $n^4$, using the quantum Weyl algebra to define the quadratic relations gives more relations than the quantum-exterior-algebra approach, yet 'seems' insufficient to make $\operatorname{Det}_q^{[m]}$ central — an observation stated without proof. And quantum analogues of Cayley's hyperdeterminants beyond the first have not been constructed in the noncommutative setting. Both bear on the conjecture, but neither is itself the object of it.

References

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