Given an algebra and a finite group acting on it via automorphisms, a natural object of study is the associated skew group algebra. In this article, we study the relationship between quasi-hereditary structures on the original algebra and on the corresponding skew group algebra. Assuming a natural compatibility condition on the partial order, we show that the skew group algebra is quasi-hereditary if and only if the original algebra is. Moreover, we show that in this setting an exact Borel subalgebra of the original algebra which is invariant as a set under the group action gives rise to an exact Borel subalgebra of the skew group algebra and that under this construction, properties such as normality and regularity of the exact Borel subalgebra are preserved.

Given a quasi-hereditary algebra , we present conditions which guarantee that the algebra obtained by grading by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that might possess. The method involves working with a pair consisting of a quasi-hereditary algebra and a (positively) graded subalgebra . The algebra arises as a quotient of by a defining ideal of . Along the way, we also show that the standard (Weyl) modules for have a structure as graded modules for . These results are applied to obtain new information about the finite dimensional algebras (e.g., the -Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated with semisimple algebraic groups in positive characteristic . These results require, at least at present, considerable restrictions on the size of .

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.