Algebraic Variants of NTRU
If one expands the discussion to include LWE-type schemes, isn't Module-LWE naturally a &quot;non-commutative&quot; hardness assumption which has practical applications? For example, the Kyber NIST PQC candidate is an MLWE scheme, and is a round 3 finalist.
In general though, I mostly see lattices over &quot;exotic&quot; (at least more so than things like RLWE/MLWE) algebraic structures discussed by coding theorists (two specific examples are Fredrique Oggier and Cong Ling). As there seems to be a larger theoretical basis of research on &quot;that side&quot; of things, it might be easier to search for practical applications.