DGA

DGAs

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Hopf algebras

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Opposite and enveloping algebras: attachments/Pasted%20image%2020220207230527.png

Properties

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Misc

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DG Lie Algebras

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A Gerstenhaber algebra is a graded \(k\)-module \(A\) together with a graded-commutative multiplication and a degree-1 Lie bracket that are compatible via the Poisson relation \begin{align*} [a, b c]=[a, b] c+(-1)^{|b|(|a|-1)} b[a, c] . \end{align*} on homogeneous elements \(a, b, c \in A\),

See BV algebra

See A_infty: attachments/Pasted%20image%2020220207231256.png attachments/Pasted%20image%2020220207231317.png

Definitions

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Minimal models for spheres

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Formal DGAs

attachments/Pasted%20image%2020220213224849.png ## Homotopy Groups of a DGA

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Formality

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Spheres

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Links to this page
  • smooth algebra

    A definition due to Kontsevich: a DGA \(A\) is smooth if \(A \in \mathsf{Perf}\left(A \otimes A^{^{\operatorname{op}}} \right)\). It is compact if \(\operatorname{dim} H^{\bullet}(A, d)<\infty\). This properties are preserved under the derived Morita equivalence.

  • perfect complexes

    Idea: an analog in complexes of the notion of a finite dimensional vector space (finiteness and dualizability). One can relate \(\mathsf{Perf}(X)\) for \(X\) a separated scheme of finite type with \({\mathrm{perf}}(A)\) for \(A\) a single DGA, and if \(X\) is a smooth scheme then \(A\) is a smooth algebra.

  • algebra over a ring
  • THH
    This offers two different perspectives on associative algebras: homological invariants are “abelian” (i. e. arise when one works with an additive category, e.g. chain complexes of modules over a ring), while homotopical invariants are “non-abelian” (i. e. arise from non-additive categories, like the category of all DGAs).
  • Maurer-Cartan
  • Griffiths and Morgan Reading Notes

    Overall purpose: want to relate \(C^\infty\) forms on a manifold to AT invariants. One significant result: given a manifold \(M\), the singular cohomology \(H^*(M, {\mathbf{R}})\) is isomorphic to the cohomology of the differential graded algebra of \(C^\infty\) forms, \(H^*_{DR}(M)\).

  • Bousfield localization

    Relation to the minimal model program and DGAs: attachments/Pasted%20image%2020220403210151.png Relation to completion: attachments/Pasted%20image%2020220403210311.png

  • A_infty

    Idea: associativity only up to higher coherent homotopy. Naturally present on the homology complex of any DGA.

#AG/deformation-theory #todo/add-references ##