  We generalize our recent explicit construction of the full hierarchy of
Baxter Q-operators of compact spin chains with su(n) symmetry to the
supersymmetric case su(n|m). The method is based on novel degenerate solutions
of the graded Yang-Baxter equation, leading to an amalgam of bosonic and
fermionic oscillator algebras. Our approach is fully algebraic, and leads to
the exact solution of the associated compact spin chains while avoiding Bethe
ansatz techniques. It furthermore elucidates the algebraic and combinatorial
structures underlying the system of nested Bethe equations. Finally, our
construction naturally reproduces the representation, due to Z. Tsuboi, of the
hierarchy of Baxter Q-operators in terms of hypercubic Hasse diagrams.
