Commutativity of quantum elementary symmetric polynomials in half-quantum algebras

In half-quantum algebras $\mathcal{H}_\epsilon(R,F)$ — obtained from a quantum matrix algebra associated with a compatible pair of braidings $(R,F)$ by imposing only half of its defining relations — the quantum determinant and the quantum elementary symmetric polynomials $e_k$ are well defined, but, as observed by Gurevich and Saponov (2021), the available relations are insufficient to prove that the $e_k$ commute with each other, whereas the more restrictive current relations of the associated generalized (braided) Yangians do imply commutativity and yield a Bethe subalgebra. The problem is to determine the commutativity status of the $e_k$ in half-quantum algebras — to prove that $[e_k,e_l]=0$ follows from the half-system alone, or to exhibit a half-quantum algebra and a representation in which two elementary symmetric polynomials fail to commute — thereby clarifying the precise algebraic input needed for commuting integrals of motion.

Background

Let $V$ be a finite-dimensional vector space over a field of characteristic 0. A braiding (or $R$-matrix) on $V$ is an invertible operator $R\in\operatorname{End}(V^{\otimes 2})$ satisfying the braid form of the Yang–Baxter equation $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$. A braiding is a Hecke symmetry if $(R-q\,I)(R+q^{-1}I)=0$ for an invertible parameter $q$, and an involutive symmetry if $R^2=I$. A Hecke symmetry admits the spectral decomposition $R=q\,S^{(2)}-q^{-1}A^{(2)}$, where $S^{(2)}$ and $A^{(2)}$ are the $q$-symmetrizer and $q$-antisymmetrizer; these projectors extend to $k$-fold tensor factors as $S^{(k)}$ and $A^{(k)}$. A braiding is called skew-invertible if there exists an operator $\Psi\in\operatorname{End}(V^{\otimes 2})$ such that $\operatorname{Tr}_2(R_{12}\Psi_{23})=P_{13}$, where $P_{13}$ is the flip of the first and third tensor factors; such a $\Psi$ makes the categorical trace (the $R$-trace, $\operatorname{Tr}_R X=\operatorname{Tr}(C^R X)$ with $C^R=\operatorname{Tr}_2\Psi$) available. A braiding is even of rank $m$ if the corresponding $R$-exterior algebra $\Lambda_R(V)$ has finitely many nonzero homogeneous components and its top component $\Lambda_R^m(V)$ is one-dimensional.

Two braidings $R$ and $F$ on $V$ form a compatible pair if $R_{12}F_{23}F_{12}=F_{23}F_{12}R_{23}$ and $F_{12}F_{23}R_{12}=R_{23}F_{12}F_{23}$. Given a compatible pair, the quantum matrix algebra $\mathcal{L}(R,F)$ is the unital algebra generated by the entries of a matrix $L=\|l_{ji}\|_{1\le i,j\le N}$, $N=\dim V$, subject to the quadratic relations

$$R_{12}\bar L_1\bar L_2=\bar L_1\bar L_2R_{12}\tag{2.3}$$

written in tensor form, where $\bar L_1=L_1=L\otimes I$ acts on the first factor of $V\otimes V$ and $\bar L_2=F_{12}L_1F_{12}^{-1}$ is obtained from it by conjugation by $F$. Two fundamental cases are $F=P$ (the flip, i.e. the operator swapping the two tensor factors), giving the RTT algebra (the Reshetikhin–Takhtajan–Faddeev algebra of quantized functions on a matrix group), and $F=R$, giving the reflection equation (RE) algebra. A key structural fact (Gurevich–Saponov 2021, Proposition 5) is that the defining system (2.3) is equivalent to the union of two half-systems:

$$S^{(2)}\bar L_1\bar L_2A^{(2)}=0\quad\Longleftrightarrow\quad \bar L_1\bar L_2A^{(2)}=A^{(2)}\bar L_1\bar L_2A^{(2)},\tag{3.5}$$ $$A^{(2)}\bar L_1\bar L_2S^{(2)}=0\quad\Longleftrightarrow\quad A^{(2)}\bar L_1\bar L_2=A^{(2)}\bar L_1\bar L_2A^{(2)}.\tag{3.6}$$

Imposing only one of the two half-systems defines the right and left half-quantum algebras $\mathcal{H}_r(R,F)$ and $\mathcal{H}_\ell(R,F)$; each is strictly larger than $\mathcal{L}(R,F)$, which is a quotient of it. Half-quantum-type matrices were introduced by Isaev and Ogievetsky (2013) as matrices satisfying the single relation $S^{(2)}\bar M_1\bar M_2A^{(2)}=0$ (with antecedents in Manin's work, as the survey notes in Remark 6); their particular cases include the Manin matrices — noncommutative analogues of matrices with commutative entries — and their $q$-deformations studied by Chervov–Falqui–Rubtsov–Silantyev. For half-quantum matrices the Cayley–Hamilton–Newton theorem holds: each matrix over the algebra satisfies its own characteristic polynomial. In a half-quantum algebra one can also define, via the higher $R$-antisymmetrizers $A^{(k)}$, the quantum elementary symmetric polynomials

$$e_k=\operatorname{Tr}_{R(1\dots k)}(A^{(k)}\bar L_1\cdots\bar L_k),\quad 1\le k\le m,$$

where $\operatorname{Tr}_{R(1\dots k)}=\operatorname{Tr}_{R(1)}\cdots\operatorname{Tr}_{R(k)}$ is the $R$-trace over the first $k$ tensor factors; the source survey notes that in an HQA the projector $A^{(k)}$ can equivalently be transferred to the rightmost position or placed on both sides of the chain of $\bar L$-matrices, and that the quantum determinant differs from the top polynomial $e_m$ only by a factor.

For the full quantum matrix algebras the analogous symmetric functions are known to commute: in the RE algebra the symmetric functions of the quantum eigenvalues form the characteristic subalgebra, which is commutative and central (Gurevich–Pyatov–Saponov 1997). For the generalized (braided) Yangians $Y(R,F)$ — analogues of the Drinfeld Yangian (a Hopf deformation of the enveloping algebra of a current algebra), generated by the entries of a matrix $L(u)$ whose entries are formal series in a spectral parameter — the defining current relations read

$$R_{12}(u,v)\bar L_1(u)\bar L_2(v)=\bar L_1(v)\bar L_2(u)R_{12}(u,v),\tag{4.5}$$

where $R(u,v)$ is a rational or trigonometric current braiding (a spectral-parameter-dependent braiding satisfying the parameterized Yang–Baxter equation). In $Y(R,F)$ the same polynomials $e_k$, taken at all values of the spectral parameter, generate the Bethe subalgebra $\mathcal{B}(R,F)$ — so named because its elements form a commuting family of integrals of motion (quantum Hamiltonians) of the associated integrable model — and Gurevich–Saponov–Slinkin (2020) proved that $\mathcal{B}(R,F)$ is commutative for any compatible pair with $R$ a skew-invertible involutive or Hecke symmetry. The proof uses the full current relations. At the half-points $u-v=1$ (rational case) and $u/v=q^2$ (trigonometric case) the current braiding becomes proportional to the antisymmetrizer $A^{(2)}$, and the Yangian relations reduce to

$$A^{(2)}\bar L_1(u)\bar L_2(u-1)=\bar L_1(u-1)\bar L_2(u)A^{(2)}\tag{4.6}$$

in the rational case, and to

$$A^{(2)}\bar L_1(u)\bar L_2(q^{-2}u)=\bar L_1(q^{-2}u)\bar L_2(u)A^{(2)}\tag{4.7}$$

in the trigonometric case — that is, to HQA-type relations imposed at all parameter shifts; the survey credits these more restrictive relations, as compared with the defining relations of an HQA, with making the commutativity property establishable in the generalized Yangians.

In the half-quantum algebra itself the situation is open. Gurevich and Saponov (2021) state verbatim that “in any HQA the number of relations between the generators is insufficient to prove that the elementary polynomials commute with each other”, and conclude that “it is not possible to prove the commutativity of the quantum elementary polynomials in a HQA”, whereas in the generalized Yangians the more restrictive relations (4.6) and (4.7) do enable this property. Isaev and Ogievetsky (2013) had earlier asserted, without proof, that — contrary to the algebra generated by the full quantum matrix — the sets of elementary and complete symmetric functions of half-quantum matrices are not commutative. Neither a proof of commutativity in half-quantum algebras nor a rigorous counterexample is known.

Problem Statement

Let $(R,F)$ be a compatible pair of braidings on a finite-dimensional vector space $V$, where $R$ is a skew-invertible Hecke symmetry with $q^2\ne1$ (equivalently, a Hecke symmetry that is not involutive), even of rank $m$. Let $\mathcal{H}_\epsilon(R,F)$, $\epsilon\in\{r,\ell\}$, denote the half-quantum algebra generated by the entries of the matrix $L=\|l_{ji}\|_{1\le i,j\le N}$, $N=\dim V$, subject only to the half-system (3.5) (right) or (3.6) (left), and let $e_k=\operatorname{Tr}_{R(1\dots k)}(A^{(k)}\bar L_1\cdots\bar L_k)$, $1\le k\le m$, be the quantum elementary symmetric polynomials defined through the $R$-antisymmetrizers and the $R$-trace, the displayed formula serving as the definition of $e_k$ in both $\mathcal{H}_r(R,F)$ and $\mathcal{H}_\ell(R,F)$. Determine, for each $\epsilon\in\{r,\ell\}$, the commutativity status of the family $\{e_k\}_{1\le k\le m}$ in $\mathcal{H}_\epsilon(R,F)$: either exhibit a compatible pair $(R,F)$ with the properties above, a choice of $\epsilon\in\{r,\ell\}$, and an explicit representation (or a certified normal form) of the corresponding half-quantum algebra in which $[e_k,e_l]\ne 0$ for some $k,l$ — a single counterexample, for either value of $\epsilon$, settles the question, since it establishes that $[e_k,e_l]=0$ is not a consequence of the half-system alone within the stated class — or prove that no such instance exists within the stated class, i.e. that $[e_k,e_l]=0$ is a consequence of the half-system alone for every admissible pair $(R,F)$ and every $\epsilon\in\{r,\ell\}$.

References

1. D. Gurevich, P. Saponov, “Determinants in quantum matrix algebras and integrable systems”, Theoret. Math. Phys. 207:2 (2021) 626–639. DOI 10.1134/S004057792105007X. English translation of Teor. Mat. Fiz. 207:2 (2021) 261–276, DOI 10.4213/tmf10043. Preprint arXiv:1906.07287, https://arxiv.org/abs/1906.07287.
2. A. Isaev, O. Ogievetsky, “Half-quantum linear algebra”, in: Symmetries and Groups in Contemporary Physics (Nankai Ser. Pure Appl. Math. Theoret. Phys. 11), World Scientific, 2013, 479–486. DOI 10.1142/9789814518550_0066. Preprint arXiv:1303.3991, https://arxiv.org/abs/1303.3991.
3. A. Isaev, O. Ogievetsky, P. Pyatov, “On quantum matrix algebras satisfying the Cayley–Hamilton–Newton identities”, J. Phys. A: Math. Gen. 32 (1999) L115–L121. DOI 10.1088/0305-4470/32/9/002, https://doi.org/10.1088/0305-4470/32/9/002.
4. D. Gurevich, P. Saponov, A. Slinkin, “Bethe subalgebras in braided Yangians and Gaudin-type models”, Commun. Math. Phys. 374:2 (2020) 689–704. DOI 10.1007/s00220-019-03514-4. Preprint arXiv:1810.03126, https://arxiv.org/abs/1810.03126.
5. D. Gurevich, P. Saponov, “Braided Yangians”, J. Geom. Phys. 138 (2019) 124–143. DOI 10.1016/j.geomphys.2018.12.024. Preprint arXiv:1612.05929, https://arxiv.org/abs/1612.05929.
6. D. Gurevich, P. Pyatov, P. Saponov, “Hecke symmetries and characteristic relations on reflection equation algebras”, Lett. Math. Phys. 41 (1997) 255–264. DOI 10.1023/A:1007386006326. Preprint arXiv:q-alg/9605048, https://arxiv.org/abs/q-alg/9605048.
7. A. Chervov, G. Falqui, V. Rubtsov, “Algebraic properties of Manin matrices I”, Adv. Appl. Math. 43 (2009) 239–315. DOI 10.1016/j.aam.2009.02.003. Preprint arXiv:0901.0235, https://arxiv.org/abs/0901.0235.
8. A. Chervov, G. Falqui, V. Rubtsov, A. Silantyev, “Algebraic properties of Manin matrices II: q-analogues and integrable systems”, Adv. Appl. Math. 60 (2014) 25–89. DOI 10.1016/j.aam.2014.06.001. Preprint arXiv:1210.3529, https://arxiv.org/abs/1210.3529.
9. A. Silantyev, “Manin Matrices for Quadratic Algebras”, SIGMA 17 (2021) 066. DOI 10.3842/SIGMA.2021.066, https://doi.org/10.3842/SIGMA.2021.066. Preprint arXiv:2009.05993, https://arxiv.org/abs/2009.05993.
10. R.-Q. Jian, “A quantum shuffle approach to quantum determinants”, Lett. Math. Phys. 112:6 (2022) Paper 127. DOI 10.1007/s11005-022-01615-1. Preprint arXiv:2112.02518, https://arxiv.org/abs/2112.02518.
11. D. Gurevich, V. Petrova, P. Saponov, “Matrix Capelli identities related to reflection equation algebra”, J. Geom. Phys. 179 (2022) 104606. DOI 10.1016/j.geomphys.2022.104606. Preprint arXiv:2207.02034, https://arxiv.org/abs/2207.02034.
12. D. V. Talalaev, “RE-algebras, quasi-determinants and the full Toda system”, Partial Differ. Equ. Appl. Math. 11 (2024) 100898. DOI 10.1016/j.padiff.2024.100898. Preprint arXiv:2406.07434, https://arxiv.org/abs/2406.07434.
13. S. Kožić, “On the h-adic quantum vertex algebras associated with Hecke symmetries”, Commun. Math. Phys. 397:2 (2023) 607–634. DOI 10.1007/s00220-022-04498-4. Preprint arXiv:2202.08190, https://arxiv.org/abs/2202.08190.
14. D. Gurevich, P. Saponov, “Quantum Schur-Weyl duality and q-Frobenius formula related to reflection equation algebras”, Int. Math. Res. Not. 2025:3 (2025) rnae288. DOI 10.1093/imrn/rnae288. Preprint arXiv:2307.06677, https://arxiv.org/abs/2307.06677.
15. D. Gurevich, P. Saponov, “Generalized Harish-Chandra morphism on reflection equation algebras”, J. Geom. Phys. 210 (2025) 105435. DOI 10.1016/j.geomphys.2025.105435. Preprint arXiv:2411.01303, https://arxiv.org/abs/2411.01303.
16. D. Gurevich, P. Saponov, “Braided Gelfand-Zetlin algebras and their semiclassical counterparts”, Physica D 483 (2025) 134940. DOI 10.1016/j.physd.2025.134940. Preprint arXiv:2507.04564, https://arxiv.org/abs/2507.04564.
17. D. Gurevich, P. Saponov, M. Zaitsev, “Wick theorem and matrix Capelli identity for quantum differential operators on Reflection Equation Algebras”, preprint arXiv:2412.13373 (2024), https://arxiv.org/abs/2412.13373.
18. D. Gurevich, P. Saponov, M. Zaitsev, “Representations of Spectrum of GL(m) type Quantum Matrices”, preprint arXiv:2601.11306 (2026), https://arxiv.org/abs/2601.11306.