Hochschild cohomology of the quadratic monomial algebra \rm N_m

2024/03/29 by Itagaki, Tomohiro, Nakamoto, Kazunori, Torii, Takeshi
#18G40 #FOS: Mathematics #Primary 16E40 #Rings and Algebras (math.RA) #Secondary 16S37

paper · doi:10.48550/arxiv.2403.20074

Abstract

Let \rm N_m(R) = \ (a_ij) ∈ \rm M_m(R) | a₁₁ = a₂₂ = ⋯ = a_mm and a_ij = 0 for any i > j \ for a commutative ring R. Then \rm N_m(R) is a quadratic monomial algebra over R. We calculate \rm HH^∗(\rm N_m(R), \rm M_m(R)/\rm N_m(R)) as R-modules. We also determine the R-algebra structure of the Hochschild cohomology ring \rm HH^∗(\rm N_m(R), \rm N_m(R)). For m ≥ 3, \rm HH^∗(\rm N_m(R), \rm N_m(R)) is an infinitely generated algebra over R and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket.

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