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Kevin De Laet

Publications and source records attributed to Kevin De Laet.

13 recordsLinked to original sources
arXiv2019

4-dimensional Artin-Schelter regular quadratic $\tilde{H}_4$-algebras

In this paper, quadratic algebras on which $\tilde{H}_4$, the Heisenberg group of order 64, acts as degree-preserving algebra automorphisms are studied. In particular, we show that if $\mathcal{A}$ is a four-dimensional Artin-Schelter regular quadratic $\tilde{H}_4$-algebra with the degree one part isomorphic to the Schrödinger representation of $\tilde{H}_4$, then $\mathcal{A}$ is (a twist of) a four-dimensional Sklyanin algebra or (a twist of) a quantum Clifford algebra of global dimension 4.

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arXiv2018

Representation Theory of $\mathbb{Z}_2^{*n}$

We study the representations of the group $\mathbb{Z}_2^{*n}$, the free product of $\mathbb{Z}_2$ with itself $n$-times. We use the action of $B_n = S_2 \wr S_n $ as algebra automorphisms on the group algebra $\mathbb{C}(\mathbb{Z}_2^{*n})$ to find the components that contain simple representations and to study smoothness of their GIT-quotients. In particular, all the possible local quiver settings are studied for the component containing the standard $n$-dimensional representation of $S_{n+1}$.

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arXiv2017

The irreducible representations of 3-dimensional Sklyanin algebras

In this article a complete description is given of the simple representations of a 3-dimensional Sklyanin algebra associated to a torsion point. In order to determine these irreducible representations, a review is given of classical results regarding representation theory of graded rings with excellent homological and algebraic properties.

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arXiv2016

On the center of 3-dimensional and 4-dimensional Sklyanin algebras

In this article, a new proof is given of the description of the center of quadratic Sklyanin algebras of global dimension three and four and the center of cubic Sklyanin algebras of global dimension three. The representation theory of the Heisenberg groups $H_2$, $H_3$ and $H_4$ will play an important role. In addition a new proof is given of Van den Bergh's result regarding noncommutative quadrics.

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arXiv2016

Constructing $G$-algebras

In this article we define $G$-algebras, that is, graded algebras on which a reductive group $G$ acts as gradation preserving automorphisms. Starting from a finite dimensional $G$-module $V$ and the polynomial ring $\mathbb{C}[V]$, it is shown how one constructs a sequence of projective varieties $\mathbf{V}_k$ such that each point of $\mathbf{V}_k$ corresponds to a graded algebra with the same decomposition up to degree $k$ as a $G$-module. After some general theory, we apply this to the case that $V$ is the $n+1$-dimensional permutation representation of $S_{n+1}$, the permutation group on $n+1$ letters.

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arXiv2015

Quotients of degenerate Sklyanin algebras

In this paper it is shown how the Heisenberg group of order 27 can be used to construct quotients of degenerate Sklyanin algebras. These quotients have properties similar to the classical Sklyanin case in the sense that they have the same Hilbert series, the same character series and a central element of degree 3. Regarding the central element of a 3-dimensional Sklyanin algebra, a better way to view this using Heisenberg-invariants is shown.

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arXiv2015

The point variety of quantum polynomial rings

We show that the reduced point variety of a quantum polynomial algebra is the union of specific linear subspaces in $\mathbb{P}^n$, we describe its irreducible components and give a combinatorial description of the possible configurations in small dimensions.

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arXiv2015

Character series and Sklyanin algebras at points of order 2

This paper has two goals: to prove certain properties of character series of graded algebras on which a finite group acts as algebra automorphisms and to provide a detailed analysis of representations of 5-dimensional Sklyanin algebras at points of order 2. We also prove that for any odd prime p, the p-dimensional Sklyanin algebras associated to points of order 2 are graded Clifford algebras.

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arXiv2014

Geometry of representations of quantum spaces

The quantum plane $A=\mathbb{C}_ρ[x,y,z]$ with $ρ$ a root of unity has singularities in its representation variety $\mathbf{trep}_nA$ and its center $Z(A)$. Using the technique of a noncommutative blow-up, we prove that this technique fails in contrast to the 3-dimensional Sklyanin algebras if we want to resolve the singularities in the representation variety. However, we will see that the singularity of the center in the origin can be made better using this technique.

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arXiv2014

Graded Clifford algebras of prime global dimension with an action of H_p

In this article we study graded Clifford algebras with a gradation preserving action of automorphisms given by $H_p$, the Heisenberg group of order $p^3$ with $p$ prime. After reviewing results in dimensions 3 and 4, we will determine the graded Clifford algebras that are AS-regular algebras of global dimension 5 and generalize certain results to arbitrary dimension $p$. More specifically, we prove that there is a nice correspondence between $\mathbb{P}^1_{\mathbb{F}_p}$ and the algebras isomorphic to the quantum space $\mathbb{C}_{-1}[x_0,\ldots,x_{p-1}]$.

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arXiv2014

The geometry of representations of 3-dimensional Sklyanin algebras

The representation scheme ${\tt rep}_n A$ of the 3-dimensional Sklyanin algebra $A$ associated to a plane elliptic curve and n-torsion point contains singularities over the augmentation ideal $\mathfrak{m}$. We investigate the semi-stable representations of the noncommutative blow-up algebra $B=A \oplus \mathfrak{m}t \oplus\mathfrak{m}^2 t^2 \oplus ...$ to obtain a partial resolution of the central singularity ${\tt proj} Z(B) \rightarrow {\tt spec} Z(A)$ such that the remaining singularities in the exceptional fiber determine an elliptic curve and are all of type $\mathbb{C} \times \mathbb{C}^2/\mathbb{Z}_n$.

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