scholarly journals The Homotopy Lie Algebra of a Complex Hyperplane Arrangement Is Not Necessarily Finitely Presented

2008 ◽  
Vol 17 (2) ◽  
pp. 129-143 ◽  
Author(s):  
Jan-Erik Roos
Keyword(s):  
Lie Algebra ◽  
Hyperplane Arrangement ◽  
Finitely Presented ◽  
Complex Hyperplane

Poincaré series of Milnor algebras and free arrangements

2010 ◽  
Vol 47 (4) ◽  
pp. 513-521
Author(s):  
Shahid Ahmad
Keyword(s):  
Hyperplane Arrangement ◽  
Betti Numbers ◽  
Poincaré Series ◽  
Poincare Series ◽  
Complex Hyperplane

We show that for a free complex hyperplane arrangement \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} $$\mathcal{A}$$ \end{document}: f = 0, the Poincaré series of the graded Milnor algebra M(f) and the Betti numbers of the arrangement complement M(\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} $$\mathcal{A}$$ \end{document}) determine each other. Examples show that this is false if we drop the freeness assumption.

scholarly journals Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

2012 ◽  
Vol 206 ◽  
pp. 75-97 ◽  
Author(s):  
Alexandru Dimca
Keyword(s):  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Monodromy Operator ◽  
Low Dimensions ◽  
Milnor Fiber ◽  
Necessary And Sufficient ◽  
Complex Hyperplane

AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

Cubature rules from Hall–Littlewood polynomials

2020 ◽  
Author(s):  
J F van Diejen ◽  
E Emsiz
Keyword(s):  
Haar Measure ◽  
Symplectic Group ◽  
Symmetric Functions ◽  
Hyperplane Arrangement ◽  
Cubature Formulas ◽  
Littlewood Polynomials ◽  
Orthogonality Relations ◽  
Unitary Ensemble ◽  
Determinantal Expression ◽  
Complex Hyperplane

Abstract Discrete orthogonality relations for Hall–Littlewood polynomials are employed so as to derive cubature rules for the integration of homogeneous symmetric functions with respect to the density of the circular unitary ensemble (which originates from the Haar measure on the special unitary group $SU(n;\mathbb{C})$). By passing to Macdonald’s hyperoctahedral Hall–Littlewood polynomials, we moreover find analogous cubature rules for the integration with respect to the density of the circular quaternion ensemble (which originates in turn from the Haar measure on the compact symplectic group $Sp (n;\mathbb{H})$). The cubature formulas under consideration are exact for a class of rational symmetric functions with simple poles supported on a prescribed complex hyperplane arrangement. In the planar situations (corresponding to $SU(3;\mathbb{C})$ and $Sp (2;\mathbb{H})$), a determinantal expression for the Christoffel weights enables us to write down compact cubature rules for the integration over the equilateral triangle and the isosceles right triangle, respectively.

scholarly journals Circle-valued Morse theory for complex hyperplane arrangements

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Toshitake Kohno ◽  
Andrei Pajitnov
Keyword(s):  
Morse Theory ◽  
Hyperplane Arrangement ◽  
Hyperplane Arrangements ◽  
Complex Hyperplane

AbstractLet 𝒜 be an essential complex hyperplane arrangement in

scholarly journals Finite presentability of some metabelian Hopf algebras

2005 ◽  
Vol 72 (1) ◽  
pp. 109-127 ◽  
Author(s):  
Dessislava H. Kochloukova
Keyword(s):  
Abelian Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Abelian Ideal ◽  
Finite Presentability ◽  
Finitely Presented ◽  
Homological Type

We classify the Hopf algebras U (L)#kQ of homological type FP2 where L is a Lie algebra and Q an Abelian group such that L has an Abelian ideal A invariant under the Q-action via conjugation and U (L/A)#kQ is commutative. This generalises the classification of finitely presented metabelian Lie algebras given by J. Groves and R. Bryant.

scholarly journals An algorithm for analysis of the structure of finitely presented Lie algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Vladimir V. Kornyak
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nonlinear Partial Differential Equations ◽  
Practical Importance ◽  
Physical Models ◽  
Generators And Relations ◽  
Defining Relations ◽  
Number Of Generators ◽  
Finite Set ◽  
Finitely Presented

International audience We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.

scholarly journals $\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

2017 ◽  
Vol 2018 (7) ◽  
pp. 2070-2098 ◽  
Author(s):  
Misha V Feigin ◽  
Alexander P Veselov
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Special Class ◽  
Vector Fields ◽  
Hyperplane Arrangement ◽  
Gradient Vector ◽  
Coxeter Systems ◽  
Flat Coordinates

Abstract It is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated with $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.

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