  We show that the category of graded modules over a finite-dimensional graded
algebra admitting a triangular decomposition can be endowed with the structure
of a highest weight category. When the algebra is self-injective, we show
furthermore that this highest weight category has tilting modules in the sense
of Ringel. This provides a new perspective on the representation theory of such
algebras, and leads to several new structures attached to them. There are a
wide variety of examples in algebraic Lie theory to which this applies:
restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras,
finite quantum groups, and restricted rational Cherednik algebras.
