  Building up on our previous works regarding $q$-deformed $P$-partitions, we
introduce a new family of subalgebras for the ring of quasisymmetric functions.
Each of these subalgebras admits as a basis a $q$-analogue to Gessel's
fundamental quasisymmetric functions where $q$ is equal to a complex root of
unity. Interestingly, the basis elements are indexed by sets corresponding to
an intermediary statistic between peak and descent sets of permutations that we
call extended peak.
