Hochschild cohomology for free semigroup algebras
Linzhe Huang, Minghui Ma, Xiaomin Wei

TL;DR
This paper investigates the Hochschild cohomology of free semigroup operator algebras, establishing automatic continuity of derivations and computing the vanishing of certain cohomology groups, thus advancing the understanding of their algebraic structure.
Contribution
It introduces a new computational method for Hochschild cohomology and proves the triviality of first and higher cohomology groups for key free semigroup algebras.
Findings
All derivations of these algebras are automatically continuous.
The first Hochschild cohomology group of _{\u03bb} with coefficients in _{\u03bbl} is zero.
Higher cohomology groups of non-commutative disc algebras vanish when |bb|<init.
Abstract
This paper focuses on the cohomology of operator algebras associated with the free semigroup generated by the set , with the left regular free semigroup algebra and the non-commutative disc algebra serving as two typical examples. We establish that all derivations of these algebras are automatically continuous. By introducing a novel computational approach, we demonstrate that the first Hochschild cohomology group of with coefficients in is zero. Utilizing the Ces\`aro operators and conditional expectations, we show that the first normal cohomology group of is trivial. Finally, we prove that the higher cohomology groups of the non-commutative disc algebras with coefficients in the complex field vanish when .…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Operator Algebra Research · Homotopy and Cohomology in Algebraic Topology
