A Computation of Tamarkin-Tsygan Calculus
Jun Chen, Xiabing Ruan, Jia Yang
TL;DR
This paper provides a complete computation of the Tamarkin–Tsygan calculus for the Koszul algebra $A = \mathbf{k}\langle x,y,z\rangle/( x^{2}+yx,xz,zy )$, whose global dimension is $4$ while having only $3$ generators. Using the Koszul resolution and algebraic Morse theory to relate it to the two-sided bar resolution, the authors obtain explicit bases for Hochschild homology and cohomology, verify vanishing in high degrees, and derive all TT-operations: cup product, cap product, Connes' differential, and Gerstenhaber bracket. Cup products are largely trivial and Connes' differential admits compact, explicit formulas; in contrast, cap products and Gerstenhaber brackets are highly intricate, with detailed, case-by-case formulas. The results illuminate the nontrivial geometry of TT-calculus in a noncommutative setting and showcase the power of resolution- and Morse-theoretic methods to render explicit, verifiable structures in Hochschild (co)homology. Overall, the work provides a concrete, richly structured example guiding future intuition about noncommutative TT-calculus and its invariants.
Abstract
We compute the full Tamarkin-Tsygan calculus of a Koszul algebra whose global dimension exceeds the number of generators. Our results show that even for algebras possessing an economic presentation and agreeable homological properties, the Hochschild (co)homology, as well as the structure of the Tamarkin--Tsygan calculus may exhibit a rather intricate behavior.