A Geometric Approach to Hochschild Cohomology of the Exterior Algebra

2016/07/09 by Michael Wong, Wong, Michael
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1607.02661

openalex publication_date 2016/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a new computation of Hochschild (co)homology of the exterior algebra, together with algebraic structures, by direct comparison with the symmetric algebra. The Hochschild cohomology is determined to be essentially the algebra of even-weight polyvector fields. From Kontsevich's formality theorem, the differential graded Lie algebra of Hochschild cochains is proved to be formal when the vector space generating the exterior algebra is even dimensional. We conjecture that formality fails in the odd dimensional case, proving this when the dimension is one. In all dimensions, formal deformations of the exterior algebra are classified by formal Poisson structures.

Citations

• Hochschild (co)homology and Koszul duality
• Another proof of M. Kontsevich formality theorem
• Operads, homotopy algebra and iterated integrals for double loop spaces
• Minimal Resolutions of Algebras
• Hochschild cohomology of group extensions of quantum complete intersections
• The Hochschild cohomology ring of a Frobenius algebra with semisimple\n Nakayama automorphism is a Batalin-Vilkovisky algebra

Related

• Hochschild (co)homology of exterior algebras
• On the Hochschild (co)homology of quantum exterior algebras
• The Hochschild cohomology ring of the singular cochain algebra of a space
• Hochschild cohomology of Lie-Rinehart algebras
• On the A_∞-Formality conjecture