Hochschild cohomology and homology of quantum complete intersections

Oppermann, Steffen

doi:10.2140/ant.2010.4.821

Cited by 19 publications

(12 citation statements)

References 7 publications

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“…Also [α, γ] ∈ HH n−1 odd ( V ), which is trivial unless n = 1. This statement in particular recovers the results regarding the additive and cup product structure in [8], [5], [30], [36] and [27], as well as the results about the Gerstenhaber structure in [19] and [18]. Remark 5.6.…”

Section: But This Precisely Equalssupporting

confidence: 82%

“…Despite the ubiquity of the exterior algebra in mathematics, many of its Hochschild structures are still not fully understood or have resisted simple interpretation. As for what is known, computations of the Hochschild (co)homology groups of the exterior algebra and the cohomological ring structure have been given in various contexts : as a special case of the results for quantum complete intersections ( [8], [5], [30]), from direct combinatorial techniques [36], and from algebraic Morse theory [27]. Recently, the Gerstenhaber bracket has been described for quantum complete intersections and their group extensions ( [19], [18]), building off the work of [6] on twisted tensor products and of [29] on defining the bracket on complexes other than the bar complex.…”

Section: Introductionmentioning

confidence: 99%

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A Geometric Approach to Hochschild Cohomology of the Exterior Algebra

Wong

2016

Preprint

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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. Contents1. Introduction 2.1. Hochschild complexes 2.2. Koszul algebras 2.3. The symmetric and exterior algebras 3. Hochschild homology 4. Hochschild cohomology 5. Algebraic structure on Hochschild cohomology 5.1. Gerstenhaber structure 5.2. BV structure 6. Formality theorems and deformation theory Appendix A. Hochschild (co)homology of a graded algebra References

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Section: But This Precisely Equalssupporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

A Geometric Approach to Hochschild Cohomology of the Exterior Algebra

Wong

2016

Preprint

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“…, 0). This grading induces an internal grading on Ext n A c q (k, k) for all n, and from the paragraph preceding [Opp,Corollary 3.5] it follows that the internal degree of z i in Ext 2 [Opp,Corollary 3.5], for each 1 i c, every element in Ext 2 …”

Section: Varietiesmentioning

confidence: 98%

The Avrunin–Scott theorem for quantum complete intersections

Bergh¹,

Erdmann²

2009

Journal of Algebra

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“…Then the algebra A q = k x 1 , ..., x n / (x mi i , x i x j − q ij x j x i ) is known a quantum complete intersection. Many authors have studied the Hochschild cohomology of quantum complete intersections; for example [12,13,28,46]. In [50] the authors proved in many cases that the first Hochschild cohomology is a solvable Lie algebra.…”

Section: 2mentioning

confidence: 99%

Maximal tori in $HH^1$ and the fundamental group

Briggs¹,

Degrassi²

2021

Preprint

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We investigate maximal tori in the Hochschild cohomology Lie algebra HH 1 (A) of a finite dimensional algebra A, and their connection with the fundamental groups associated to presentations of A. We prove that every maximal torus in HH 1 (A) arises as the dual of some fundamental group of A, extending work of Farkas, Green and Marcos; de la Peña and Saorín; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of A is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.

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Hochschild cohomology and homology of quantum complete intersections

Abstract: We compute the Hochschild cohomology and homology for arbitrary finitedimensional quantum complete intersections. It turns out that their behavior varies widely, depending on the choice of commutation parameters, and we give precise criteria for when to expect what behavior.

Cited by 19 publications

References 7 publications

A Geometric Approach to Hochschild Cohomology of the Exterior Algebra

A Geometric Approach to Hochschild Cohomology of the Exterior Algebra

The Avrunin–Scott theorem for quantum complete intersections

Maximal tori in $HH^1$ and the fundamental group

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