arXiv subjects

S. Paul Smith

Publications and source records attributed to S. Paul Smith.

At least 19 recordsLinked to original sources

arXiv2023

The symplectic leaves for the elliptic Poisson bracket on projective space defined by Feigin-Odesskii and Polishchuk

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 $E$.

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arXiv2021

Modular properties of elliptic algebras

Fix a pair of relatively prime integers $n>k\ge 1$, and a point $(η\,|\,τ)\in\mathbb{C}\times\mathbb{H}$, where $\mathbb{H}$ denotes the upper-half complex plane, and let ${{a\;\,b}\choose{c\,\;d}}\in\mathrm{SL}(2,\mathbb{Z})$. We show that Feigin and Odesskii's elliptic algebras $Q_{n,k}(η\,|\,τ)$ have the property $Q_{n,k}\big(\fracη{cτ+d}\,\big\vert\,\frac{aτ+b}{cτ+d}\big)\cong Q_{n,k}(η\,|\,τ)$. As a consequence, given a pair $(E,ξ)$ consisting of a complex elliptic curve $E$ and a point $ξ\in E$, one may unambiguously define $Q_{n,k}(E,ξ):=Q_{n,k}(η\,|\,τ)$ where $τ\in\mathbb{H}$ is any point such that $\mathbb{C}/\mathbb{Z}+\mathbb{Z}τ\cong E$ and $η\in\mathbb{C}$ is any point whose image in $E$ is $ξ$. This justifies Feigin and Odesskii's notation $Q_{n,k}(E,ξ)$ for their algebras.

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arXiv2021

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

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

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arXiv2020

Feigin and Odesskii's elliptic algebras

We study the elliptic algebras $Q_{n,k}(E,τ)$ introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers $n>k\geq 1$, an elliptic curve $E$, and a point $τ\in E$. We consider and compare several different definitions of the algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we show that $Q_{n,k}(E,0)$, and $Q_{n,n-1}(E,τ)$ are polynomial rings on $n$ variables. We also show that $Q_{n,k}(E,τ+ζ)$ is a twist of $Q_{n,k}(E,τ)$ when $ζ$ is an $n$-torsion point. This paper is the first of several we are writing about the algebras $Q_{n,k}(E,τ)$.

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arXiv2020

The characteristic variety for Feigin and Odesskii's elliptic algebras

This paper examines an algebraic variety that controls an important part of the structure and representation theory of the algebra $Q_{n,k}(E,τ)$ introduced by Feigin and Odesskii. The $Q_{n,k}(E,τ)$'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 $τ\in E$. It is already known that the structure and representation theory of $Q_{n,1}(E,τ)$ 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 $z\mapsto z+τ$ interacts with that geometry. For $k\ge 2$ a similar phenomenon occurs: $(E,τ)$ is replaced by $(X_{n/k},σ)$ where $X_{n/k}\subseteq\mathbb P^{n-1}$ is the characteristic variety of the title and $σ$ is an automorphism of it that is determined by the negative continued fraction for $\frac{n}{k}$. There is a surjective morphism $Φ:E^g \to X_{n/k}$ 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 $Φ$ 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 $Q_{n,k}(E,τ)$. The backward $σ$-orbit of each point on $X_{n/k}$ determines a point module for $Q_{n,k}(E,τ)$.

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arXiv2020

Finite quotients of powers of an elliptic curve

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.

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arXiv2020

Elliptic R-matrices and Feigin and Odesskii's elliptic 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 $n>k\ge 1$, a complex elliptic curve $E$, and a point $τ\in E$. 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 $τ$, it 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 $R_τ(z):\mathbb{C}^n\otimes\mathbb{C}^n\to\mathbb{C}^n\otimes\mathbb{C}^n$, $z\in\mathbb{C}$, that (we will show) satisfy the quantum Yang-Baxter equation with spectral parameter.

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arXiv2020

New Artin-Schelter regular and Calabi-Yau algebras via normal extensions

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.

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arXiv2018

A geometric invariant of $6$-dimensional subspaces of $4\times 4$ matrices

Let $k$ be an algebraically closed field and ${\sf G}(2,k^4)$ the Grassmannian of 2-planes in $k^4$. We associate to each 6-dimensional subspace $R$ of the space of 4x4 matrices over $k$ a closed subscheme ${\bf X}_R \subseteq {\sf G}(2,k^4)$. We show that each irreducible component of ${\bf X}_R$ has dimension at least one and when ${\rm dim}({\bf X}_R)=1$, then ${\rm deg}({\bf X}_R)=20$ where degree is computed with respect to the ambient ${\mathbb P}^5$ under the Plücker embedding ${\sf G}(2,k^4) \to {\mathbb P}^5$. We give two examples involving elliptic curves: in one case ${\bf X}_R$ is the secant variety for a quartic elliptic curve, so ${\rm dim}({\bf X}_R)=2$, in the other ${\bf X}_R$ is a curve having 7 irreducible components, three of which are elliptic curves, and four of which are smooth conics.

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arXiv2018

Simple modules over the 4-dimensional Sklyanin Algebras at points of finite order

In 1982 E.K. Sklyanin defined a family of graded algebras $A(E,τ)$, depending on an elliptic curve $E$ and a point $τ\in E$ 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 $\le n$ and that "almost all" have dimension precisely $n$. There are enough finite dimensional simple modules to separate elements of $A$; that is, if $0\ne a \in A$, then there exists a simple module $S$ such that $a.S \ne 0.$ 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, ${\mathcal A}$ say, of finite ${\mathcal O}_S$ algebras where $S$ is the projective 3-fold determined by the center of $A$. We determine where ${\mathcal A}$ is Azumaya, and prove that the division algebra ${\rm Fract}({\mathcal A})$ has rational center. Thus, for each $E$ and each $τ\in E$ of order $n \ne 0,2,4$ one obtains a division algebra of degree $s$ over the rational function field of ${\mathbb P}^3$, where $s=n$ if $n$ is odd, and $s={{1} \over {2}} n$ 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.

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arXiv2017

Non-commutative Geometry of Homogenized Quantum $\mathfrak{sl}(2,\mathbb{C})$

This paper examines the relationship between certain non-commutative analogues of projective 3-space, $\mathbb{P}^3$, and the quantized enveloping algebras $U_q(\mathfrak{sl}_2)$. The relationship is mediated by certain non-commutative graded algebras $S$, one for each $q \in \mathbb{C}^\times$, having a degree-two central element $c$ such that $S[c^{-1}]_0 \cong U_q(\mathfrak{sl}_2)$. The non-commutative analogues of $\mathbb{P}^3$ are the spaces $\operatorname{Proj}_{nc}(S)$. We show how the points, fat points, lines, and quadrics, in $\operatorname{Proj}_{nc}(S)$, and their incidence relations, correspond to finite dimensional irreducible representations of $U_q(\mathfrak{sl}_2)$, Verma modules, annihilators of Verma modules, and homomorphisms between them.

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arXiv2017

Some algebras having relations like those for the 4-dimensional Sklyanin algebras

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.

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arXiv2016

Exotic Elliptic Algebras of dimension 4 (with an Appendix by Derek Tomlin)

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.

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arXiv2016

The Classification of 3-Calabi-Yau algebras with 3 generators and 3 quadratic relations

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 $TV/(R)$ the quotient of the tensor algebra on $V$ by the ideal generated by $R$. Raf Bocklandt proved that if $TV/(R)$ is 3-Calabi-Yau, then it is isomorphic to $J({\sf{w}})$, the "Jacobian algebra" of some ${\sf{w}} \in V^{\otimes 3}$. This paper classifies the ${\sf{w}}\in V^{\otimes 3}$ such that $J({\sf{w}})$ is 3-Calabi-Yau. The classification depends on how ${\sf{w}}$ transforms under the action of the symmetric group $S_3$ on $V^{\otimes 3}$ and on the nature of the subscheme $\{\overline{\sf{w}}=0\} \subseteq \mathbb{P}^2$ where $\overline{\sf{w}}$ denotes the image of ${\sf{w}}$ in the symmetric algebra $SV$. Surprisingly, as ${\sf{w}}$ ranges over $V^{\otimes 3}-\{0\}$, only nine isomorphism classes of algebras appear as non-3-Calabi-Yau $J({\sf{w}})$'s.

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arXiv2016

Corrigendum to "Maps between non-commutative spaces" [Trans. Amer. Math. Soc., 356(7) (2004) 2927-2944]

The statement of Lemma 3.1 in the published paper 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 $\mathbb{N}$-graded $k$-algebra such that ${\rm dim}_k(A_n)< \infty$ for all $n$, and $J$ a graded two-sided ideal of $A$. If the non-commutative scheme ${\sf Proj}_{nc}(A)$ is isomorphic to a projective scheme $X$, then there is a closed subscheme $Z \subseteq X$ such that ${\sf Proj}_{nc}(A/J)$ is isomorphic to $Z$. This result is a geometric translation of what we actually prove: if the category ${\sf QGr}(A)$ is equivalent to ${\sf Qcoh}(X)$, then ${\sf QGr}(A/J)$ is equivalent to ${\rm Qcoh}(Z)$ for some closed subscheme $Z \subseteq X$.

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arXiv2015

Exotic Elliptic Algebras

This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "exotic elliptic algebra". We show that the twisted algebra has many of the good properties that the Sklyanin algebra has, and that it has some new properties that make it quite unusual by comparison.

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arXiv2015

m-Koszul Artin-Schelter regular algebras

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian $m$-Koszul twisted Calabi-Yau or, equivalently, $m$-Koszul Artin-Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w,i) for a unique-up-to-scalar-multiples twisted superpotential w in a tensor power of some vector space V. By definition, D(w,i) is the quotient of the tensor algebra TV by the ideal generated by all i-th order left partial derivatives of w. We identify the group of graded algebra automorphisms of D(w,i) with a subgroup of GL(V). We show that the homological determinant of a graded algebra automorphism $σ$ of an $m$-Koszul Artin-Schelter regular algebra D(w,i) is the scalar hdet($σ$) given by the formula hdet($σ$) w =$σ^{\otimes m+i}$(w). It follows from this that the homological determinant of the Nakayama automorphism of an $m$-Koszul Artin-Schelter regular algebra is 1. As an application, we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian Artin-Schelter regular algebras of dimension 3.

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arXiv2014

The Grothendieck group of non-commutative non-noetherian analogues of $\mathbb{P}^1$ and regular algebras of global dimension two

Let $V$ be a finite-dimensional positively-graded vector space. Let $b \in V \otimes V$ be a homogeneous element whose rank is $\text{dim}(V)$. Let $A=TV/(b)$, the quotient of the tensor algebra $TV$ modulo the 2-sided ideal generated by $b$. Let ${\sf gr}(A)$ be the category of finitely presented graded left $A$-modules and ${\sf fdim}(A)$ its full subcategory of finite dimensional modules. Let ${\sf qgr}(A)$ be the quotient category ${\sf gr}(A)/{\sf fdim}(A)$. We compute the Grothendieck group $K_0({\sf qgr}(A))$. In particular, if the reciprocal of the Hilbert series of $A$, which is a polynomial, is irreducible, then $K_0({\sf qgr}(A)) \cong \mathbb{Z}[θ] \subset \mathbb{R}$ as ordered abelian groups where $θ$ is the smallest positive real root of that polynomial. When $\text{dim}_k(V)=2$, ${\sf qgr}(A)$ is equivalent to the category of coherent sheaves on the projective line, $\mathbb{P}^1$, or a stacky $\mathbb{P}^1$ if $V$ is not concentrated in degree 1. If $\text{dim}_k(V) \ge 3$, results of Piontkovskii and Minamoto suggest that ${\sf qgr}(A)$ behaves as if it is the category of "coherent sheaves" on a non-commutative, non-noetherian, analogue of $\mathbb{P}^1$.

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