scholarly journals Non-abelian cohomology jump loci from an analytic viewpoint

2014 ◽  
Vol 16 (04) ◽  
pp. 1350025 ◽  
Author(s):  
Alexandru Dimca ◽  
Ştefan Papadima
Keyword(s):  
Minimal Model ◽  
Nilpotent Groups ◽  
Exponential Map ◽  
Flat Connections ◽  
Differential Graded Algebra ◽  
Irreducible Components ◽  
Rank One ◽  
Positive Weights ◽  
Projective Manifolds ◽  
Representation Varieties

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

scholarly journals RANK TWO TOPOLOGICAL AND INFINITESIMAL EMBEDDED JUMP LOCI OF QUASI-PROJECTIVE MANIFOLDS

2018 ◽  
Vol 19 (2) ◽  
pp. 451-485 ◽  
Author(s):  
Stefan Papadima ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Linear Algebraic Group ◽  
The Other ◽  
Fundamental Groups ◽  
Flat Connections ◽  
Other Hand ◽  
Projective Manifolds ◽  
Representation Varieties

We study the germs at the origin of $G$-representation varieties and the degree 1 cohomology jump loci of fundamental groups of quasi-projective manifolds. Using the Morgan–Dupont model associated to a convenient compactification of such a manifold, we relate these germs to those of their infinitesimal counterparts, defined in terms of flat connections on those models. When the linear algebraic group $G$ is either $\text{SL}_{2}(\mathbb{C})$ or its standard Borel subgroup and the depth of the jump locus is 1, this dictionary works perfectly, allowing us to describe in this way explicit irreducible decompositions for the germs of these embedded jump loci. On the other hand, if either $G=\text{SL}_{n}(\mathbb{C})$ for some $n\geqslant 3$, or the depth is greater than 1, then certain natural inclusions of germs are strict.

scholarly journals Homology TQFT's and the Alexander–Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

2003 ◽  
Vol 55 (4) ◽  
pp. 766-821 ◽  
Author(s):  
Thomas Kerler
Keyword(s):  
Moduli Spaces ◽  
Hopf Algebras ◽  
Large Family ◽  
Quantum Invariants ◽  
Topological Quantum ◽  
Irreducible Components ◽  
Skein Theory ◽  
Higher Rank ◽  
The One ◽  
Representation Varieties

AbstractWe develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology ofU(1)-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra= ℤ/2 n ⋊ Λ* ℝ2on the other side. We find that both TQFT's are SL(2; ℝ)-equivariant functors and, as such, are isomorphic. The SL(2; ℝ)-action in the Hennings construction comes from the natural action onand in the case of the Frohman–Nicas theory from the Hard–Lefschetz decomposition of theU(1)-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of SL(2; ℤ) and Sp(2g; ℤ), thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg–Witten theories, Casson type theories for homology circlesà laDonaldson, higher rank gauge theories following Frohman and Nicas, and the ℤ=pℤ reductions of Reshetikhin.Turaev theories over the cyclotomic integers ℤ[ζp]. We also conjecture that the Hennings TQFT for quantum-sl2is the product of the Reshetikhin–Turaev TQFT and such a homological TQFT.

scholarly journals Automorphisms of nilpotent groups of class two with small rank

1985 ◽  
Vol 39 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Thomas A. Fournelle
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Direct Sums ◽  
Matrix Computations ◽  
Central Factor ◽  
Rank One ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractRational abelian groups, that is, torsion-free abelian groups of rank one, are characterized by their types. This paper characterizes rational nilpotent groups of class two, that is, nilpotent groups of class two in which the center and central factor group are direct sums of rational abelian groups. This characterization is done according to the types of the summands of the center and the central factor group. Using these types and some cohomological techniques it is possible to determine the automorphism group of the nilpotent group in question by performing essentially matrix computations.In particular, the automorphism groups of rational nilpotent groups of class two and rank three are completely described. Specific examples are given of semicomplete and pseudocomplete nilpotent groups.

Minimal Free Multi-Models for Chain Algebras

2004 ◽  
Vol 11 (4) ◽  
pp. 733-752 ◽  
Author(s):  
J. Huebschmann
Keyword(s):  
Perturbation Theory ◽  
Local Ring ◽  
Minimal Model ◽  
Graded Algebra ◽  
Differential Graded Algebra ◽  
Homological Perturbation Theory ◽  
Homological Perturbation

Abstract Let 𝑅 be a local ring and 𝐴 a connected differential graded algebra over 𝑅 which is free as a graded 𝑅-module. Using homological perturbation theory techniques, we construct a minimal free multi-model for 𝐴 having properties similar to those of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi’ refers to the category of multicomplexes.

scholarly journals REPRESENTATION VARIETIES OF ALGEBRAS WITH NODES

2021 ◽  
pp. 1-31
Author(s):  
Ryan Kinser ◽  
András C. Lőrincz
Keyword(s):  
Moduli Spaces ◽  
Rational Singularities ◽  
Orbit Closures ◽  
Node Splitting ◽  
Irreducible Components ◽  
Representation Varieties

Abstract We study the behaviour of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the ‘relative setting’ (splitting one node at a time), we demonstrate that there are many nonhereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show that this is true when irreducible components are replaced by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras.

scholarly journals Smoothable del Pezzo surfaces with quotient singularities

2009 ◽  
Vol 146 (1) ◽  
pp. 169-192 ◽  
Author(s):  
Paul Hacking ◽  
Yuri Prokhorov
Keyword(s):  
Minimal Model ◽  
Del Pezzo Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Rank One ◽  
Quotient Singularities ◽  
Del Pezzo ◽  
Moduli Problems

AbstractWe classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the 3-fold minimal model program and also in moduli problems.

scholarly journals ALGEBRAIC GEOMETRIC INVARIANTS OF PARAFREE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 155-169 ◽  
Author(s):  
SAL LIRIANO
Keyword(s):  
Algebraic Variety ◽  
Free Groups ◽  
Central Sequence ◽  
Geometric Invariants ◽  
Number Of Generators ◽  
Irreducible Components ◽  
One Relator Groups ◽  
The Difference ◽  
Representation Varieties ◽  
Trivial Consequence

Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL2ℂ inherits the structure of an algebraic variety known as the representation variety of G in SL2ℂ. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that are not free share a host of properties with free groups. In this paper algebraic geometric invariants involving the number of maximal irreducible components (mirc) of R(G), and the dimension of R(G) for certain classes of parafree groups are computed. It is shown that in an infinite number of cases these invariants successfully discriminate between ismorphism types within the class of parafree groups of the same rank. This is quite surprising, since an n generated group G is free of rank n if and only if Dim (R(G)) = 3n. In fact, a trivial consequence of Theorem 1.8 in this paper is that given an arbitrary positive integer k, and any integer r ≥ 2, there exist infinitely many non-isomorphic fg parafree groups of rank r and deviation 1 with representation varieties of dimension 3r, having more than k mirc of dimension 3r. This paper also introduces many novel and surprising dimension formulas for the representation varieties of certain one-relator groups.

scholarly journals Flat connections and resonance varieties: From rank one to higher ranks

2016 ◽  
Vol 369 (2) ◽  
pp. 1309-1343 ◽  
Author(s):  
Daniela Anca Măcinic ◽  
Ştefan Papadima ◽  
Clement Radu Popescu ◽  
Alexander I. Suciu
Keyword(s):  
Flat Connections ◽  
Rank One ◽  
Resonance Varieties

scholarly journals POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

2019 ◽  
Vol 62 (3) ◽  
pp. 706-736
Author(s):  
DAVID KYED ◽  
HENRIK DENSING PETERSEN
Keyword(s):  
Lie Group ◽  
Nilpotent Groups ◽  
Group Cohomology ◽  
Exponential Map ◽  
Nilpotent Lie Group ◽  
Polynomial Maps ◽  
Polynomial Coefficients ◽  
Classical Polynomials ◽  
Simply Connected ◽  
Refined Version

AbstractWe introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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