On finite dimensional 2d fractional superalgebras

The formal definition of heterotic fractional supersymmetries (FS) (Q−1/k)k=P−1 is studied. Viewing FS as a finite dimensional subalgebras of the ZF and Tye et al. FSC invariances, we propose more rigorous definitions of these subalgebras. The Z3 and Z4 ZF parafermionic symmetries and the Level k=4 Tye et al. ones are studied in details. Known results such as the Leclair and Vafa fractional superalgebras are obtained as special cases.