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To $B$ we also associate a (quantum) lattice Heisenberg algebra $\\mathfrak{h}_B$. We show that, provided $B$ is not concentrated in degree zero, the Grothendieck group of $\\mathcal{H}_B$ is isomorphic, as an algebra, to $\\mathfrak{h}_B$. For specific choices of Frobenius algebra $B$, we recover existing results, including those of Khovanov and Cautis--Licata. 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