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 but forgotten until it was rediscovered by Philip Hall (1959), both of whom published no more than brief summaries of their work. The Hall polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of Masaki Kashiwara and George Lusztig regarding canonical bases in quantum groups. generalized Hall algebras to more general categories, such as the category of representations of a quiver.
Construction.
A finite abelian "p"-group "M" is a direct sum of cyclic "p"-power components formula_1 where
formula_2 is a partition of formula_3 called the "type" of "M". Let formula_4 be the number of subgroups "N" of "M" such that "N" has type formula_5 and the quotient "M/N" has type formula_6. Hall proved that the functions "g" are polynomial functions of "p" with integer coefficients. Thus we may replace "p" with an indeterminate "q", which results in the Hall polynomials 
 formula_7
Hall next constructs an associative ring formula_8 over formula_9, now called the Hall algebra. This ring has a basis consisting of the symbols formula_10 and the structure constants of the multiplication in this basis are given by the Hall polynomials: 
formula_11
It turns out that "H" is a commutative ring, freely generated by the elements formula_12 corresponding to the elementary "p"-groups. The linear map from "H" to the algebra of symmetric functions defined on the generators by the formula
 formula_13
(where "e""n" is the "n"th elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements formula_10 may be interpreted via the Hall–Littlewood symmetric functions. Specializing "q" to 0, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.