Hall algebra

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 E. 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 polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of Kashiwara–Lusztig's 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 ${\displaystyle C_{p^{\lambda _{i}}},}$ where ${\displaystyle \lambda =(\lambda _{1},\lambda _{2},\ldots )}$ is a partition of ${\displaystyle n}$ called the type of M. Let ${\displaystyle g_{\mu ,\nu }^{\lambda }(p)}$ be the number of subgroups N of M such that N has type ${\displaystyle \nu }$ and the quotient M/N has type ${\displaystyle \mu }$. 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

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