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Poisson geometry and representations of PI 4-dimensional Sklyanin algebras

Abstract

Take S to be a 4-dimensional Sklyanin (elliptic) algebra that is module-finite over its center Z; thus, S is PI. Our first result is the construction of a Poisson Z-order structure on S such that the induced Poisson bracket on Z is non-vanishing. We also provide the explicit Jacobian structure of this bracket, leading to a description of the symplectic core decomposition of the maximal spectrum Y of Z. We then classify the irreducible representations of S by combining (1) the geometry of the Poisson order structures, with (2) algebro-geometric methods for the elliptic curve attached to S, along with (3) representation-theoretic methods using line and fat point modules of S. Along the way, we improve results of Smith and Tate obtaining a description the singular locus of Y for such S. The classification results for irreducible representations are in turn used to determine the zero sets of the discriminants ideals of these algebras S.

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Acknowledgements

The authors thank S. Paul Smith for generously making available his 1993 work [36] via private communication and for also making it available for public use recently on the ArXiv e-print service. The authors also thank the anonymous referee for their numerous comments, which enabled us to greatly improve the quality of this manuscript.

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Correspondence to Milen Yakimov.

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Walton and Yakimov are partially supported by NSF Grants #1663775, 1601862, 1901830, 1903192, 2100756, and 2131243. Walton is also supported by the Sloan Foundation. Wang is partially supported by Simon collaboration Grant #688403.

Index of Notation

Index of Notation

a, 12 \(h_1,h_2\), 11 \(\rho \), 11
\(a_1, a_2\), 14 \(h_{{\overline{z}}}\), 17 \(\rho _1, \rho _2, \rho _3\), 14
\({{\,\mathrm{{\mathrm {ad}} }\,}}_r\), 33   
\(\alpha \), 8 \(\iota \), 26, S, 4, 8
\(\widetilde{\alpha }\), 28   \(S'\), 33
  \(J({\underline{F}})_{k,l}\), 37 \(S(\alpha ,\beta ,\gamma )\), 8
B, 9   \(S_\hbar , \widehat{S}_\hbar \), 28
\(B_{\hbar }, \widehat{B}_{\hbar }\), 29 \(S_{[\kappa _1:\kappa _2]}\), 39 \(S_{[\kappa _1:\kappa _2]}\), 5, 39
b, 12   s, 10
\(\beta \), 8 \(L_\hbar \), 29 \(\sigma \), 8
\(\widetilde{\beta }\) ,28 \({\mathcal {L}}\), 9 \(\sigma _{\hbar }\), 29
  \({\mathcal {L}}_{\hbar }\), 29  
\(C(\omega +k\tau )\), 19 \({\mathcal {L}}_i\), 9 T, 35
\({\mathcal {C}}(\omega +k\tau )\), 45 \(\ell _1, \ell _2\), 11 \(T^{[k]}\), 35
c, 12 \(\ell _{p,q}\), 16 \(\tau \), 8
  \(\Lambda \), 13 \(\theta \), 26
\(\delta \), 33   \(\theta _B\), 29
\({{\,\mathrm{{\mathrm {Der}} }\,}}(A/C)\), 26 M(p), 43 \(\theta _R\), 30
  M(pq), 16 \(\theta _S\), 28
\(E, {\widehat{E}}\), 8 \({\mathfrak {m}}_y\), 5, 40  
\(E'\), 11 N, 26 \(u_0,\ldots ,u_3\), 11
\(E''\), 10 \(N_4\), 12  
\({\overline{E}}\), 17 n, 4, 9 \(V(\omega +k\tau )\), 6, 43
\(E_2\), 12   \(v_0, \ldots , v_3\), 8
\({\overline{E}}_2\), 17 \(\Omega _k\), 35 \(v^{'}_0, \ldots , v^{''}_3\), 29
\(E_{\hbar }\), 29 \(\Omega (z)\), 16  
\(e_0, e_1, e_2, e_3\), 8 \(\omega \), 12 w, 29
\(\overline{\epsilon }_i\), 17 \(\overline{\omega }_i\), 17  
\(\epsilon \), \(\epsilon _1\), \(\epsilon _2\), 12   x,33
  \(\overline{{\mathbb {P}}}\), 17 \(x_0, \ldots , x_3\),8
\(F_1\), \(F_2\), 11 \(\mathcal {P}(I)\), 27 \(\widetilde{x}_0, \ldots , \widetilde{x}_3\), 31
\(F(\omega +k\tau )\), 43 \(\partial \), \(\partial _z\), 26 \(\xi \), 12
\(f_1, f_2\), 11 \(\Phi _1\), \(\Phi _2\), 11  
\(f_{{\overline{z}}}\), 17 \(\phi _1\), \(\phi _2\), 8 Y, 4, 16
  \(\pi \), 17 \(Y_0^{symp}\), 4, 16
G, 17 \(\psi , \psi _1, \psi _2\), 33 \(Y^{sing}\), 4, 16
g, 17 \(\psi _k\), 35 \(Y_{\gamma _1}, Y_{\gamma _2}\), 4, 16
\(g'\), 39 \(\psi ', \psi ''\), 39 \((Y_{\gamma _1, \gamma _2})^{sing}\), 4, 16
\(g_1\), \(g_2\), 9 \(\psi _{\hbar }\), 29 \((Y^{sing})_{\gamma _1,\gamma _2}\), 4, 16
\(\widetilde{g_1}, \widetilde{g_2}\), 28   \(Y_{[\kappa _1:\kappa _2]}\), 40
\(\gamma \), 8 \(Q({\overline{z}})\), 17  
\(\widetilde{\gamma }\), 28   Z, 4, 11
  \(R_\hbar \), 29 \(z_0, \ldots , z_3\), 11
\(H_4\), 12 \(r_i\), 17 \(\widetilde{z}_0, \ldots , \widetilde{z}_3\), 31
\({\hbar }\), 28   \(\overline{z}\), 17

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Walton, C., Wang, X. & Yakimov, M. Poisson geometry and representations of PI 4-dimensional Sklyanin algebras. Sel. Math. New Ser. 27, 99 (2021). https://doi.org/10.1007/s00029-021-00713-7

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Keywords

  • 4-dimensional Sklyanin algebra
  • Poisson order
  • Azumaya locus
  • Singular locus
  • Irreducible representation

Mathematics Subject Classification

  • 14A22
  • 16G99
  • 17B63
  • 81S10