This paper determines the symplectic leaves for a remarkable Poisson structure on $\mathbb{C}\mathbb{P}^{n-1}$ discovered by Feigin and Odesskii, and, independently, by Polishchuk. The Poisson bracket is determined by a holomorphic line bundle of degree $n \ge 3$ on a compact Riemann surface of genus one or, equivalently, by an elliptic normal curve $E\subseteq\mathbb{C}\mathbb{P}^{n-1}$. The symplectic leaves are described in terms of higher secant varieties to $E$.

This paper examines an algebraic variety that controls an important part of the structure and representation theory of the algebra $Q_{n,k}(E,\tau)$ introduced by Feigin and Odesskii. The $Q_{n,k}(E,\tau)$'s are a family of quadratic algebras depending on a pair of coprime integers $n>k\ge 1$, an elliptic curve $E$, and a point $\tau\in E$. It is already known that the structure and representation theory of $Q_{n,1}(E,\tau)$ is controlled by the geometry associated to $E$ embedded as a degree $n$ normal curve in the projective space $\mathbb P^{n-1}$, and by the way in which the translation automorphism $z\mapsto z+\tau$ interacts with that geometry. For $k\ge 2$ a similar phenomenon occurs: $(E,\tau)$ is replaced by $(X_{n/k},\sigma)$ where $X_{n/k}\subseteq\mathbb P^{n-1}$ is the characteristic variety of the title and $\sigma$ is an automorphism of it that is determined by the negative continued fraction for $\frac{n}{k}$. There is a surjective morphism $\Phi:E^g \to X_{n/k}$ where $g$ is the length of that continued fraction. The main result in this paper is that $X_{n/k}$ is a quotient of $E^g$ by the action of an explicit finite group. We also prove some assertions made by Feigin and Odesskii. The morphism $\Phi$ is the natural one associated to a particular invertible sheaf $\mathcal L_{n/k}$ on $E^g$. The generalized Fourier-Mukai transform associated to $\mathcal L_{n/k}$ sends the set of isomorphism classes of degree-zero invertible $\mathcal O_E$-modules to the set of isomorphism classes of indecomposable locally free $\mathcal O_E$-modules of rank $k$ and degree $n$. Thus $X_{n/k}$ has an importance independent of the role it plays in relation to $Q_{n,k}(E,\tau)$. The backward $\sigma$-orbit of each point on $X_{n/k}$ determines a point module for $Q_{n,k}(E,\tau)$.

Fix a pair of relatively prime integers n > k ≥ 1 n>k\ge 1 , and a point ( η | τ ) ∈ C × H (\eta \,|\,\tau )\in \mathbb {C}\times \mathbb {H} , where H \mathbb {H} denotes the upper-half complex plane, and let ( a b c d ) ∈ S L ( 2 , Z ) {{a\;\,b}\choose {c\,\;d}}\in \mathrm {SL}(2,\mathbb {Z}) . We show that Feigin and Odesskii’s elliptic algebras Q n , k ( η | τ ) Q_{n,k}(\eta \,|\,\tau ) have the property Q n , k ( η c τ + d | a τ + b c τ + d ) ≅ Q n , k ( η | τ ) Q_{n,k}\big (\frac {\eta }{c\tau +d}\,\big \vert \,\frac {a\tau +b}{c\tau +d}\big )\cong Q_{n,k}(\eta \,|\,\tau ) . As a consequence, given a pair ( E , ξ ) (E,\xi ) consisting of a complex elliptic curve E E and a point ξ ∈ E \xi \in E , one may unambiguously define Q n , k ( E , ξ ) ≔ Q n , k ( η | τ ) Q_{n,k}(E,\xi )≔Q_{n,k}(\eta \,|\,\tau ) where τ ∈ H \tau \in \mathbb {H} is any point such that C / Z + Z τ ≅ E \mathbb {C}/\mathbb {Z}+\mathbb {Z}\tau \cong E and η ∈ C \eta \in \mathbb {C} is any point whose image in E E is ξ \xi . This justifies Feigin and Odesskii’s notation Q n , k ( E , ξ ) Q_{n,k}(E,\xi ) for their algebras. The algebras Q n , 1 ( η | τ ) Q_{n,1}(\eta \,|\,\tau ) are commonly known as Sklyanin algebras in honor of Sklyanin’s discovery of Q 4 , 1 ( η | τ ) Q_{4,1}(\eta \,|\,\tau ) .

The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E,tau), that depend on a quartic elliptic curve E in P^3 and a translation automorphism tau of E. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates apart from the minor difference that they are not commutative. They are elliptic analogues of the enveloping algebra of sl(2,C) and the quantized enveloping algebras U_q(gl_2). Recently, Cho, Hong, and Lau, conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by the graph of a bijection between two 20-point subsets of the projective space P^3. The paper also examines a 3-parameter family of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.

The algebras Q_n,k(E,τ ) 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≥ 1 , a complex elliptic curve E, and a point τ∈ E . The main result in this paper is that Q_n,k(E,τ ) has the same Hilbert series as the polynomial ring on n variables when τ is not a torsion point. We also show that Q_n,k(E,τ ) is a Koszul algebra, hence of global dimension n when τ is not a torsion point, and, for all but countably many τ , Q_n,k(E,τ ) is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining Q_n,k(E,τ ) is the image of an operator R_τ(τ ) that belongs to a family of operators R_τ(z):ℂ^n⊗ℂ^n→ℂ^n⊗ℂ^n , z∈ℂ , that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.

The fastICA method is a popular dimension reduction technique used to reveal patterns in data. Here we show both theoretically and in practice that the approximations used in fastICA can result in patterns not being successfully recognised. We demonstrate this problem using a two-dimensional example where a clear structure is immediately visible to the naked eye, but where the projection chosen by fastICA fails to reveal this structure. This implies that care is needed when applying fastICA. We discuss how the problem arises and how it is intrinsically connected to the approximations that form the basis of the computational efficiency of fastICA.

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

0 Introduction This is a reasonably faithful account of the ve lectures I delivered at the summer course \Geometria Algebraica no Commutativa y Espacios Cuanti-cos" for graduate students, in Spain, July 25{29, 1994. The material covered was, for the most part, an abridged version of Artin and Zhang's paper 2]. Fix a eld k. Given a Z-graded k-algebra, A say, which for simplicity is assumed to be left noetherian and locally nite dimensional, its non-commutative projective scheme is deened to be the pair proj(A) := (tails(A); A); where tails(A) is the quotient category of grmod(A), the category of nitely generated graded left A-modules, modulo its full subcategory of nite dimensional modules, and A is the image of the distinguished module A A in tails(A). If A is a quotient of a commutative polynomial ring generated in degree 1, Serre 4] proved that proj(A) is isomorphic (in an obvious sense) to

Let R R denote a 6-dimensional subspace of the ring M 4 ( k ) M_4(\Bbbk ) of 4 × 4 4 \times 4 matrices over an algebraically closed field k \Bbbk . Fix a vector space isomorphism M 4 ( k ) ≅ k 4 ⊗ k 4 M_4(\Bbbk ) \cong \Bbbk ^4 \otimes \Bbbk ^4 . We associate to R R a closed subscheme X R {\mathbf X}_R of the Grassmannian of 2-dimensional subspaces of k 4 \Bbbk ^4 , where the reduced subscheme of X R {\mathbf X}_R is the set of 2-dimensional subspaces Q ⊆ k 4 Q \subseteq \Bbbk ^4 such that ( Q ⊗ k 4 ) ∩ R ≠ { 0 } (Q \otimes \Bbbk ^4) \cap R \ne \{ 0\} . Our main result is that if X R {\mathbf X}_R has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P 5 \mathbb {P}^5 via the Plücker embedding. We present several examples of X R \mathbf X_R that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, X R {\mathbf X}_R is a P 1 \mathbb {P}^1 -bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme X R \mathbf X_R appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L \mathcal {L} of a certain graded algebra determined by R R . In that context, it has been an open question for several years to describe such L \mathcal {L} of minimal dimension, i.e., those L \mathcal {L} of dimension one. Our main result implies that if dim ⁡ ( L ) = 1 \dim (\mathcal {L}) = 1 , then, as a subscheme of P 5 \mathbb {P}^5 under the Plücker embedding, deg ⁡ ( L ) = 20 \deg (\mathcal {L}) = 20 .

This is a continuation of our previous paper 1502.01744. We examine a class of non-commutative algebras A that depend on an elliptic curve and a translation automorphism of it. They may be defined in terms of the 4-dimensional Sklyanin algebra S that is associated to the same data. The algebra A has the same Hilbert series as the polynomial ring in 4 variables, and there is an associated non-commutative variety, Proj(A), that is a non-commutative analogue of P^3. The structure and representation theory of A, and the geometric properties of Proj(A) are closely related to the geometric properties of E sitting as a quartic curve in P^3. Our main results concern the classification of point modules, fat point modules, line modules, and the incidence relations between them. The line modules are parametrized by a degree 20 curve in the Grassmannian G(1,3) that is a union of 4 disjoint plane conics and 3 disjoint quartic elliptic curves that are isomorphic to E/(t) where t runs over the three 2-torsion points. A finite quantum group related to the Heisenberg group of size 4^3 acts as auto-equivalences of the category of graded A-modules and those quantum symmetries of A play a central role in our analysis.

Let E be an elliptic curve. When the symmetric group Σ_g+1 of order (g+1)! acts on E^g+1 in the natural way, the subgroup E_0^g+1, consisting of those (g+1)-tuples whose coordinates sum to zero, is stable under the action of Σ_g+1. It is isomorphic to E^g. This paper concerns the structure of the quotient variety E^g/Σ when Σ is a subgroup of Σ_g+1 generated by simple transpositions. In an earlier paper we observed that E^g/Σ is a bundle over a suitable power, E^N, with fibers that are products of projective spaces. This paper shows that E^g/Σ has an étale cover by a product of copies of E and projective spaces with an abelian Galois group.

We introduce a new method to construct 4-dimensional Artin-Schelter regular algebras as normal extensions of (not necessarily noetherian) 3-dimensional ones. The method produces large classes of new 4-dimensional Artin-Schelter regular algebras. When applied to a 3-Calabi-Yau algebra our method produces a flat family of central extensions of it that are 4-Calabi-Yau, and all 4-Calabi-Yau central extensions having the same generating set as the original 3-Calabi-Yau algebra arise in this way. Each normal extension has the same generators as the original 3-dimensional algebra, and its relations consist of all but one of the relations for the original algebra and an equal number of new relations determined by "the missing one" and a tuple of scalars satisfying some numerical conditions. We determine the Nakayama automorphisms of the 4-dimensional algebras obtained by our method and as a consequence show that their homological determinant is 1. This supports the conjecture by Mori-Smith that the homological determinant of the Nakayama automorphism is 1 for all Artin-Schelter regular connected graded algebras. Reyes-Rogalski-Zhang proved this is true in the noetherian case.

Let R denote a 6-dimensional subspace of the ring M4(k) of 4 × 4 matrices over an algebraically closed field k. Fix a vector space isomorphism M4(k) ∼= k 4 ⊗ k. We associate to R a closed subscheme XR of the Grassmannian of 2-dimensional subspaces of k , where the reduced subscheme of XR is the set of 2-dimensional subspaces Q ⊆ k 4 such that (Q ⊗ k) ∩ R 6= {0}. Our main result is that if XR has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P via the Plücker embedding. We present several examples of XR that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, XR is a P -bundle over an elliptic curve, the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme XR appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L of a certain graded algebra determined by R. In that context, it has been an open question for several years to describe such L of minimal dimension, i.e., those L of dimension one. Our main result implies that if dim(L) = 1, then, as a subscheme of P under the Plücker embedding, deg(L) = 20.

We examine the relationship between certain noncommutative analogues of projective 3-space, P-3, and the quantized enveloping algebras U-q(sl(2)). The relationship is mediated by certain noncommutative graded algebras S, one for each q is an element of C-x, having a degree-two central element c such that S[c(-1)](0) congruent to U-q(sl(2)). The noncommutative analogues of P-3 are the spaces Proj(nc)(S). We show how the points, fat points, lines, and quadrics, in Proj(nc)(S), and their incidence relations, correspond to finite-dimensional irreducible representations of U-q(sl(2)), Verma modules, annihilators of Verma modules, and homomorphisms between them.

The definition of the homological determinant in §2.2 of the published paper is not correct. Despite that, all the statements in the paper that involve properties of the homological determinant are correct.

In 1982 E.K. Sklyanin defined a family of graded algebras A(E,τ), depending on an elliptic curve E and a point τ∈ E that is not 4-torsion. The present paper is concerned with the structure of A when τ is a point of finite order, n say. It is proved that every simple A-module has dimension ≤ n and that "almost all" have dimension precisely n. There are enough finite dimensional simple modules to separate elements of A; that is, if 0 a ∈ A, then there exists a simple module S such that a.S 0. Consequently A satisfies a polynomial identity of degree 2n (and none of lower degree). Combined with results of Levasseur and Stafford it follows that A is a finite module over its center. Therefore one may associate to A a coherent sheaf, 𝒜 say, of finite 𝒪_S algebras where S is the projective 3-fold determined by the center of A. We determine where 𝒜 is Azumaya, and prove that the division algebra Fract(𝒜) has rational center. Thus, for each E and each τ∈ E of order n 0,2,4 one obtains a division algebra of degree s over the rational function field of ℙ^3, where s=n if n is odd, and s=12 n if n is even. The main technical tool in the paper is the notion of a "fat point" introduced by M. Artin. A key preliminary result is the classification of the fat points: these are parametrized by a rational 3-fold.

Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A=A(E,τ) that depends on this data: A(E,τ)=(S(E,τ)⊗M2(k))Γ where S(E,τ) is the 4-dimensional Sklyanin algebra associated to (E,τ), M2(k) is the ring of 2×2 matrices over k, and Γ is (Z/2)×(Z/2) acting in a particular way as automorphisms of S and M2(k). The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E[2] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P3 that we denote by Projnc(A). Point modules and fat point modules determine “points” in Projnc(A). Line modules determine “lines” in Projnc(A). Line modules for A are in bijection with certain lines in P(A1⁎)≅P3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G(1,3). Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plücker P5, and that deg⁡(L)=20. The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P(A1⁎). Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.

Let k be an algebraically closed field of characteristic not 2 or 3, V a 3-dimensional vector space over k, R a 3-dimensional subspace of \(V \otimes V\), and \(\textit{TV}/(R)\) the quotient of the tensor algebra on V by the ideal generated by R. Raf Bocklandt proved that if \(\textit{TV}/(R)\) is 3-Calabi–Yau, then it is isomorphic to \(J(\mathsf{w})\), the “Jacobian algebra” of some \(\mathsf{w}\in V^{\otimes 3}\). This paper classifies the \(\mathsf{w}\in V^{\otimes 3}\) such that \(J(\mathsf{w})\) is 3-Calabi–Yau. The classification depends on how \(\mathsf{w}\) transforms under the action of the symmetric group \(S_3\) on \(V^{\otimes 3}\) and on the nature of the subscheme \(\{\overline{\mathsf{w}}=0\} \subseteq {{\mathbb {P}}}^2\) where \(\overline{\mathsf{w}}\) denotes the image of \(\mathsf{w}\) in the symmetric algebra \(\textit{SV}\).

The statement of Lemma 3.1 in Maps between non-commutative spaces (Trans. Amer. Math. Soc. 356 (2004), no. 7, 2927-2944) is not correct. Lemma 3.1 is needed for the proof of Theorem 3.2. Theorem 3.2 as originally stated is true but its "proof" is not correct. Here we change the statements and proofs of Lemma 3.1 and Theorem 3.2. We also prove a new result. Let k be a field, A a left and right noetherian N-graded k-algebra such that dim(k)(A(n)) < infinity for all n, and J a graded two-sided ideal of A. If the noncommutative scheme Proj(nc)(A) is isomorphic to a projective scheme X, then there is a closed subscheme Z subset of X such that Proj(nc)(A/J) is isomorphic to Z. This result is a geometric translation of what we actually prove: if the category QGr(A) is equivalent to Qcoh(X), then QGr(A/J) is equivalent to Qcoh(Z) for some closed subscheme Z subset of X.