Absolute geometry and the Habiro topology
Lecture notes of a course on A. Smirnov's approach to the ABC-conjecture via F1-geometry and an attempt to relate this to Jim Borger's F1-geometry based on lambda-rings.
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arXiv subjects
Lieven Le Bruyn
Publications and source records attributed to Lieven Le Bruyn.
Most irreducible representations of the 3-string braid group
As the 3-string braid group B(3) and the modular group PSL(2,Z) are both of wild representation type one cannot expect a full classification of all their finite dimensional simple representations. Still, one can aim to describe 'most' irreducible representations by constructing for each d-dimensional irreducible component X of the variety iss(n,B(3)) classifying the isomorphism classes of semi-simple n-dimensional representations of B(3) an explicit minimal etale rational map A^d --> X having a Zariski dense image. Such rational dense parametrizations were obtained for all components when n < 12 in \cite{arXiv:1003.1610v1}. The aim of the present paper is to establish such parametrizations for all finite dimensions n.
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Matrix transposition and braid reversion
Matrix transposition induces an involution on the isomorphism classes of semi-simple n-dimensional representations of the three string braid group. We show that a connected component of this variety can detect braid-reversion or that the involution acts as the identity on it. We classify the fixed-point components.
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Representation stacks, D-branes and noncommutative geometry
In this note we prove that the spec(C)-points of the representation Artin-stack [rep_n R/PGL_n] of n-dimensional representations of an affine algebra R correspond to algebra morphisms R-->A where A is an Azumaya algebra of degree n over C. We connect this to the study of D-branes and Azumaya noncommutative geometry, developed by Chien-Hao Liu and Shing-Tung Yau in a series of papers from arXiv:0709.1515 to arXiv:1003.1178
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Rationality and dense families of B(3) representations
Every irreducible component of the variety of semi-simple n-dimensional representations of the modular group has a Zariski dense subset contained in the image of an etale map from a rational quotient variety of representations of a fixed quiver Q. As an application we prove that there is a unique component of 6-dimensional representations of the three string braid group B(3) able to detect braid-reversion. Further, we give explicit rational parametrizations of dense families of simple B(3)-representations in all dimensions < 12.
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(non)commutative f-un geometry
Stressing the role of dual coalgebras, we modify the definition of affine schemes over the 'field with one element'. This clarifies the appearance of Habiro-type rings in the commutative case, and, allows a natural noncommutative generalization, the study of representations of discrete groups and their profinite completions being our main motivation.
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Noncommutative geometry and dual coalgebras
In arXiv:math/0606241v2 M. Kontsevich and Y. Soibelman argue that the category of noncommutative (thin) schemes is equivalent to the category of coalgebras. We propose that under this correspondence the affine scheme of a k-algebra A is the dual coalgebra A^o and draw some consequences. In particular, we describe the dual coalgebra A^o of A in terms of the A-infinity structure on the Yoneda-space of all the simple finite dimensional A-representations.
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Braid group B_3 irreducibles, a DIY guide
This note tells you how to construct a k(n)-dimensional family of (isomorphism classes of) irreducible representations of dimension n for the three string braid group B_3, where k(n) is an admissible function of your choosing; for example take k(n) = [ n/2 ] +1 as in arXiv:0803.2778 and arXiv:0803.2785.
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Trees of semi-simple algebras
To a tree of semi-simple algebras we associate a qurve (or formally smooth algebra) S. We introduce a Zariski- and etale quiver describing the finite dimensional representations of S. In particular, we show that all quotient varieties of the etale quiver have a natural Poisson structure induced by a double Poisson algebra structure on a certain universal localization of its path algebra. Explicit calculations are included for the group algebras of the arithmetic groups (P)SL2(Z) and GL2(Z) but the methods apply as well to congruence subgroups.
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Partial Desingularizations Arising from Non-Commutative Algebras
Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial desingularization of X with classifiable remaining singularities. In dimension 3 this explains the omnipresence of conifold singularities in partial desingularizations of quotient singularities. In higher dimensions we have a small list of singularity types generalizing the role of the conifold singularity.
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Qurves and Quivers
In this paper we associate to a qurve A (formerly known as a quasi-free or formally smooth algebra) the one-quiver Q(A) and dimension vector a(A). This pair contains enough information to reconstruct for all natural numbers n the GL(n)-etale local structure of the representation scheme rep(n,A). In an appendix we indicate how one might extend this to qurves over non-algebraically closed fields. Further, we classify all finitely generated groups G such that the group algebra lG is an l-qurve. If char(l)=0 these are exactly the virtually free groups. We determine the one-quiver setting in this case and indicate how it can be used to study the finite dimensional representations of virtually free groups. As this approach also applies to fundamental algbras of graphs of separable l-algebras, we state the results in this more general setting.
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Canonical systems and non-commutative geometry
Inspired by ideas from non-commutative geometry, unions of moduli spaces of linear control systems are identified as open subsets of infinite Grassmannians.
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Three talks on noncommutative geometry@n
Notes of three talks given at the workshop 'Hilbert schemes, non-commutative algebra and the McKay correspondence' CIRM-Luminy (France) October 2003. If A is an order over a central normal affine variety X having a stability structure such that the variety of all semi-stable A-representations is a smooth variety, then the corresponding moduli space is a partial desingularization of X and we have a complete classification of the remaining singularities.
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A non-commutative topology on rep A
We extend the Zariski topology on simp A, the finite dimensional simple A-representations, to a non-commutative topology (in the sense of Fred Van Oystaeyen) on rep A, all finite dimensional A-representations, using Jordan-Holder filtrations. The non-commutativity of the topology is enforced by the order of the composition factors.
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Non-commutative covers and the modular group
We use non-commutative geometry to study the bulk of finite dimensional representations of the modular group SL(2,Z). We give specific 2n-parameter families of 6n-dimensional representations obtained from the quotient singularity C^2/Z_6.
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One quiver to rule them all
To a formally smooth algebra A we associate a quiver setting (Q,a) containing enough information to reconstruct all the local quiver settings determining the etale local structure of finite dimensional representation schemes of A, see math.AG/9904171. Conjecturally, in a (yet to be developed) non-commutative etale topology, A is locally isomorphic to an algebra B which in turn is Morita equivalent (via the dimension vector a) to the path algebra CQ of the quiver Q.
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Isolated singularities, smooth orders and Auslander regularity
In this note we prove that a smooth order satisfying the reverse geometric engineering of singularities conditions in stringtheory (in any compactifying dimension) is Auslander regular. Moreover, we classify the etale local structure of smooth orders over an isolated central singularity.
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Noncommutative Smooth Models
We determine the central simple algebras D over a functionfield K of trancendence degree two which admit a model of smooth Cayley-Hamilton algebras. This happens if and only if there is a smooth model S of K such that the ramification divisor of a maximal S-order in D is a disjoint union of smooth curves. Further, we prove that the Brauer-Severi fibration of smooth models which are in addition maximal orders is a flat morphism and determine the number of irreducible components of the fibers.
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