  Fomin and Kirillov initiated a line of research into the realization of the
cohomology and $K$-theory of generalized flag varieties $G/B$ as commutative
subalgebras of certain noncommutative algebras. This approach has several
advantages, which we discuss. This paper contains the most comprehensive result
in a series of papers related to the mentioned line of research. More
precisely, we give a model for the $T$-equivariant $K$-theory of a generalized
flag variety $K_T(G/B)$ in terms of a certain braided Hopf algebra called the
Nichols-Woronowicz algebra. Our model is based on the Chevalley-type
multiplication formula for $K_T(G/B)$ due to the first author and Postnikov;
this formula is stated using certain operators defined in terms of so-called
alcove paths (and the corresponding affine Weyl group). Our model is derived
using a type-independent and concise approach.
