scholarly journals The cohomology rings of real toric spaces and smooth real toric varieties

2021 ◽  
pp. 1-18
Author(s):  
Matthias Franz
Keyword(s):  
Toric Varieties ◽  
Graded Algebra ◽  
Cohomology Rings ◽  
Differential Graded Algebra

We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra (dga) we present is in fact an equivariant dga model, valid for arbitrary coefficients. We deduce from our description that smooth toric varieties are $\hbox{M}$ -varieties.

scholarly journals An extention of Nomizu’s Theorem –A user’s guide–

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Simple Proof ◽  
Graded Algebra ◽  
De Rham Cohomology ◽  
Differential Graded Algebra ◽  
Finite Dimensional ◽  
De Rham ◽  
Simply Connected ◽  
Complex Valued ◽  
Solvable Lie Group

AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).

scholarly journals A COMBINATORIAL DGA FOR LEGENDRIAN KNOTS FROM GENERATING FAMILIES

2013 ◽  
Vol 15 (02) ◽  
pp. 1250059 ◽  
Author(s):  
MICHAEL B. HENRY ◽  
DAN RUTHERFORD
Keyword(s):  
Standard Form ◽  
Geometric Interpretation ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Complex Sequence ◽  
Differential Graded Algebra ◽  
Legendrian Knot ◽  
Definition Of ◽  
Morse Complex ◽  
Differential Counts

For a Legendrian knot L ⊂ ℝ3, with a chosen Morse complex sequence (MCS), we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family F, we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov–Eliashberg DGA after changing coordinates by an augmentation.

scholarly journals Invariants of Legendrian Knots in Circle Bundles

2003 ◽  
Vol 05 (04) ◽  
pp. 569-627 ◽  
Author(s):  
Joshua M. Sabloff
Keyword(s):  
Contact Structure ◽  
Homology Class ◽  
Graded Algebra ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Circle Bundle ◽  
Differential Graded Algebra ◽  
Circle Bundles ◽  
Standard Contact ◽  
Definition Of

Let M be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in M. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using "Morse–Bott" techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a nontrivial homology class, but are not Legendrian isotopic.

Differential Graded Algebras

2020 ◽  
pp. 145-150
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Algebraic Model ◽  
Lie Derivative ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Rham Complex ◽  
De Rham

This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω‎(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω‎ with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.

scholarly journals The de Rham theorem for the noncommutative complex of Cenkl and Porter

2002 ◽  
Vol 30 (11) ◽  
pp. 667-696 ◽  
Author(s):  
Luis Fernando Mejias
Keyword(s):  
Finite Type ◽  
Differential Forms ◽  
Natural Isomorphism ◽  
Graded Algebra ◽  
Differential Graded Algebra ◽  
Noncommutative Version ◽  
Simplicial Set ◽  
De Rham ◽  
Poincaré Lemma

We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.

scholarly journals (q,σ,τ)-Differential Graded Algebras

Universe ◽  
2018 ◽  
Vol 4 (12) ◽  
pp. 138 ◽  
Author(s):  
Viktor Abramov ◽  
Olga Liivapuu ◽  
Abdenacer Makhlouf
Keyword(s):  
Clifford Algebra ◽  
Quantum Plane ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Generalized Clifford Algebra

We propose the notion of ( q , σ , τ ) -differential graded algebra, which generalizes the notions of ( σ , τ ) -differential graded algebra and q-differential graded algebra. We construct two examples of ( q , σ , τ ) -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a ( σ , τ ) -twisted graded q-commutator. In order to construct the second example, we introduce the notion of ( σ , τ ) -pre-cosimplicial algebra.

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.

The Weil Algebra and the Weil Model

2020 ◽  
pp. 151-160
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Total Space ◽  
Algebra Homomorphism ◽  
Graded Algebra ◽  
Algebraic Models ◽  
Weil Algebra ◽  
Differential Graded Algebra ◽  
Definite Sense ◽  
Universal Bundle

This chapter evaluates the Weil algebra and the Weil model. The Weil algebra of a Lie algebra g is a g-differential graded algebra that in a definite sense models the total space EG of a universal bundle when g is the Lie algebra of a Lie group G. The Weil algebra of the Lie algebra g and the map f is called the Weil map. The Weil map f is a graded-algebra homomorphism. The chapter then shows that the Weil algebra W(g) is a g-differential graded algebra. The chapter then looks at the cohomology of the Weil algebra; studies algebraic models for the universal bundle and the homotopy quotient; and considers the functoriality of the Weil model.

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