  Weighted KLRW algebras are diagram algebras generalizing KLR algebras. This
paper undertakes a systematic study of these algebras culminating in the
construction of homogeneous affine cellular bases in affine types A and C,
which immediately gives cellular bases for the cyclotomic quotients of these
algebras. In addition, we construct subdivision homomorphisms that relate
weighted KLRW algebras for different quivers. As an application we obtain new
results about the (cyclotomic) KLR algebras of affine type, including
(re)proving that the cyclotomic KLR algebras of type A^{(1)}_{e} and
C^{(1)}_{e} are graded cellular algebras.
