Hochschild cohomology of the quadratic monomial algebra Nₘ

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