arXiv subjects

Alex Chirvasitu

Publications and source records attributed to Alex Chirvasitu.

12 recordsLinked to original sources

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.

math.AG↗

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.

math.RA↗

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.

math.QA↗

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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arXiv2017

Flat families of point schemes for connected graded algebras

We study truncated point schemes of connected graded algebras as families over the parameter space of varying relations for the algebras, proving that the families are flat over the open dense locus where the point schemes achieve the expected (i.e. minimal) dimension. When the truncated point scheme is zero-dimensional we obtain its number of points counted with multiplicity via a Chow ring computation. This latter application in particular confirms a conjecture of Brazfield to the effect that a generic two-generator, two-relator 4-dimensional Artin-Schelter regular algebra has seventeen truncated point modules of length six.

math.RA↗

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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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.

math.QA↗

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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