In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups.
It was first discussed by Steinitz (1901) but forgotten until it was rediscovered by Philip Hall (1959), both of whom published no more than brief summaries of their work.
The Hall algebra plays an important role in the theory of Masaki Kashiwara and George Lusztig regarding canonical bases in quantum groups.
Ringel (1990) generalized Hall algebras to more general categories, such as the category of representations of a quiver.
A finite abelian p-group M is a direct sum of cyclic p-power components
Thus we may replace p with an indeterminate q, which results in the Hall polynomials Hall next constructs an associative ring
, now called the Hall algebra.
This ring has a basis consisting of the symbols
and the structure constants of the multiplication in this basis are given by the Hall polynomials: It turns out that H is a commutative ring, freely generated by the elements
The linear map from H to the algebra of symmetric functions defined on the generators by the formula (where en is the nth elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements
Specializing q to 0, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.