  We consider self-injective finite-dimensional graded algebras admitting a
triangular decomposition. In a preceding paper, we have shown that the graded
module category of such an algebra is a highest weight category and has tilting
objects in the sense of Ringel. In this paper we focus on the degree zero part
of the algebra, the core of the algebra. We show that the core captures
essentially all relevant information about the graded representation theory.
Using tilting theory, we show that the core is cellular. We then describe a
canonical construction of a highest weight cover, in the sense of Rouquier, of
this cellular algebra using a finite subquotient of the highest weight
category. Thus, beginning with a self-injective graded algebra admitting a
triangular decomposition, we canonically construct a quasi-hereditary algebra
which encodes key information, such as graded multiplicities, of the original
algebra. Our results are general and apply to a wide variety of examples,
including restricted enveloping algebras, Lusztig's small quantum groups,
hyperalgebras, finite quantum groups, and restricted rational Cherednik
algebras. We expect that the cell modules and quasi-hereditary algebras
introduced here will provide a new way of understanding these important
examples.
