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Elliptic R-matrices and Feigin and Odesskii’s elliptic algebras

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Abstract

The algebras \(Q_{n,k}(E,\tau )\) introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers \(n>k\ge 1\), a complex elliptic curve E, and a point \(\tau \in E\). The main result in this paper is that \(Q_{n,k}(E,\tau )\) has the same Hilbert series as the polynomial ring on n variables when \(\tau \) is not a torsion point. We also show that \(Q_{n,k}(E,\tau )\) is a Koszul algebra, hence of global dimension n when \(\tau \) is not a torsion point, and, for all but countably many \(\tau \), \(Q_{n,k}(E,\tau )\) is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining \(Q_{n,k}(E,\tau )\) is the image of an operator \(R_{\tau }(\tau )\) that belongs to a family of operators \(R_{\tau }(z):{\mathbb {C}}^n\otimes {\mathbb {C}}^n\rightarrow {\mathbb {C}}^n\otimes {\mathbb {C}}^n\), \(z\in {\mathbb {C}}\), that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.

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Notes

  1. By “classical” we mean the representation theory of the enveloping algebra \(U({{\mathfrak {g}}})\) and its quantization \(U_q({{\mathfrak {g}}})\), where \({{\mathfrak {g}}}\) is a finite dimensional Lie algebra over \({{\mathbb {C}}}\).

  2. An irreducible algebraic variety X over an algebraically closed field \(\Bbbk \) is rational if \(\Bbbk (X)\), its field of rational functions, is a purely transcendental extension \(\Bbbk (x_1,\ldots , x_n)\) of \(\Bbbk \); in more geometric terms, there is a non-empty open set \(U \subseteq X\) and an open set \(V \subseteq {{\mathbb {A}}}^n_{x_1,\ldots ,x_n}\) such that \(U \cong V\).

  3. This is related to the fact that Belavin’s elliptic solutions to the quantum Yang–Baxter equation with spectral parameter degenerate to trigonometric and rational solutions.

  4. Odesskii and Feigin [38] examine finite dimensional representations of \(Q_{n,k}(E,\tau )\) when \(\tau \) has finite order.

  5. The Poisson structure \(q_{n,k}\) is analogous to the Poisson structure \(\{x,y\}:=[x,y]\) on the symmetric algebra \(S({{\mathfrak {g}}})\).

  6. [40, Thm. A] shows that certain Poisson central elements for \(q_{n,1}\) are related to a higher secant variety.

  7. The symplectic leaves for the Poisson structure on the affine variety \({{\mathbb {C}}}^n={\text {Spec}}(Q_{n,k}(E,0))\) have been studied by Feigin and Odesskii in [25, 26] and [35, 36], for example.

  8. Surprisingly, the results in our earlier papers about \(Q_{n,k}(E,\tau )\) do not use this fact in an explicit way.

  9. In this paper we need an improved version of [17, Lem. 3.13]: Lemma 5.1 below shows that for each \(m \in {{\mathbb {Z}}}\) and each \(\zeta \in \frac{1}{n}\Lambda \) there is a holomorphic function \({{\mathbb {C}}}\rightarrow {\text {End}}_{{\mathbb {C}}}(V^{\otimes 2})\), \(\tau \mapsto R_{n,k,\tau }(m\tau +\zeta )\).

  10. In [37, Rmk. 4, §1], Feigin and Odesskii say there is a close connection between the \(Q_{n,k}(E,\tau )\)’s and Belavin’s elliptic solutions to the QYBE. They do not specify the connection but refer the reader to [15]; although [15, §4] concerns an algebra \({{\mathcal {R}}}^d_\eta \) that is defined in terms of \(S_k(z)\), [15] does not refer to R(z). The algebras \({{\mathcal {R}}}^d_\eta \) in [15, §4] are generated by \(n^2d\) elements whereas \(Q_{n,k}(E,\tau )\) is generated by n elements. At the end of the introduction to [37] is an equality \(A^{(d)}=Q_{n^2d,nd-1}(E,\tau )\). The algebra \(A^{(d)}\) is not defined (perhaps it is \({{\mathcal {R}}}^d_\eta \)) and there is no explanation of the equality.

  11. This implies that the dimension of \({\text {rel}}_{n,k}(E,\tau )\) is \(\genfrac(){0.0pt}1{n}{2}\), which is the first step toward proving Theorem 1.1(1).

  12. It follows from Corollary 5.9 that \(R(z)R(-z)=0=R(-z)R(z)\) if and only if \(z \in \, \pm \, \tau + \frac{1}{n}\Lambda \), and that R(z) is an isomorphism if \(z \notin \, \pm \, \tau + \frac{1}{n}\Lambda \).

  13. This implies that S and T extend to automorphisms of \(Q_{n,k}(E,\tau )\) (cf., [17, Prop. 3.23]).

  14. The \(\sum \) symbol in [46, (3.11)] should be \(\prod \), and the symbol \(\gamma _0\) in that equation denotes a non-zero scalar.

  15. The operator \(S_k(z)\) is defined in the same way as S(z) after replacing the generator h by the new generator \(h^{-k'}\).

  16. When \(k=n-1\), \(S_{k}(z)\) equals the operator S(z) in Eq. (3.1) of Richey and Tracy’s paper [46]. Some of the calculations in this proof are similar to those that produce Eqs. (3.3)–(3.12) in [46].

  17. Since \(F(z+\frac{1}{n})=e(-\frac{kr}{n})F(z)\) we may apply Lemma 2.5 to F(z) with \(\eta _1=\frac{1}{n}\), \(\eta _2=\eta \), \(a=0\), \(b=\tfrac{kr}{n}\), \(c=n^2\), and \(d= n(-\tau -\tfrac{1}{2}(1+\eta )+\tfrac{n}{2}\eta )\). Thus, \(c \eta _1 - a\eta _2 =n \) and

    $$\begin{aligned} \tfrac{1}{2}(c \eta _1^2 - a\eta _2^2 ) + (c-a)\eta _1\eta _2 + b\eta _2 - d\eta _1&\,=\, \tfrac{1}{2} +n\eta + \tfrac{kr}{n}\eta +\tau +\tfrac{1}{2}(1+\eta )-\tfrac{n}{2}\eta \\&\,=\, \tau +\tfrac{kr}{n}\eta + \tfrac{1}{2}(n+1)\eta \quad \text {modulo }\tfrac{1}{n}{{\mathbb {Z}}}+{{\mathbb {Z}}}\eta . \end{aligned}$$

  18. We adopt the following convention: a (complex) algebraic variety is a scheme over \({{\mathbb {C}}}\) that is reduced, irreducible, separated, and of finite type. An analytic variety is a (Hausdorff) analytic space that is reduced and irreducible.

  19. Alternatively, the second inequality in (5.7) follows from the first because

    $$\begin{aligned} {\text {nullity}}R_{\tau }(-\tau -\zeta )\,=\,\dim V^{\otimes 2}-{\text {rank}}R_{\tau }(-\tau -\zeta ) \, \ge \, n^{2}-\genfrac(){0.0pt}1{n+1}{2}\,=\,\genfrac(){0.0pt}1{n}{2}. \end{aligned}$$

  20. There are some other ways to extend the definition of \({\text {rel}}_{n,k}(E,\tau )\) to all \(\tau \in {{\mathbb {C}}}\); see [17, §3.3] for more discussion.

  21. What we are calling \(R^\textrm{Od}(z)\) is obtained from Odesskii’s formula for \(R_{n,k}({{\mathcal {E}}},\eta )(u-v)\) by identifying \(x_\alpha (u)\) and \(x_\alpha (v)\) with \(x_\alpha \) and setting \(v=0\) and \(u=z\).

  22. Artin–Schelter’s proof is “by computer”. De Laet’s is “by algebra”.

  23. These facts are analogues of the fact that when E is embedded in \({{\mathbb {P}}}^2\) or \({{\mathbb {P}}}^3\) as an elliptic normal curve of degree 3, or 4, respectively, it is a complete intersection. However, when E is embedded in \({{\mathbb {P}}}^{n-1}\) as an elliptic normal curve of degree \(n \ge 5\) it is not a complete intersection.

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Acknowledgements

A.C. acknowledges support through NSF grants DMS-1801011 and DMS-2001128.

R.K. was a JSPS Overseas Research Fellow, and supported by JSPS KAKENHI Grant Numbers JP16H06337, JP17K14164, JP20K14288, and JP21H04994, Leading Initiative for Excellent Young Researchers, MEXT, Japan, and Osaka Central Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849. R.K. would like to express his deep gratitude to Paul Smith for his hospitality as a host researcher during R.K.’s visit to the University of Washington.

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  1. Department of Mathematics, University at Buffalo, Buffalo, NY, 14260-2900, USA

    Alex Chirvasitu

  2. Department of Mathematics, Graduate School of Science, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi, Osaka, 558-8585, Japan

    Ryo Kanda

  3. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    S. Paul Smith

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Chirvasitu, A., Kanda, R. & Smith, S.P. Elliptic R-matrices and Feigin and Odesskii’s elliptic algebras. Sel. Math. New Ser. 29, 31 (2023). https://doi.org/10.1007/s00029-023-00827-0

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