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 , both of whom published no more than brief summaries of their work. The Hall polynomials are the
structure constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
of the Hall algebra. The Hall algebra plays an important role in the theory of
Masaki Kashiwara
is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Kashiwara made leading contributions towards algebraic analysis, microlocal analysis, D-module, ''D''-module theory, Hodge theory, sheaf theory and represent ...
and
George Lusztig regarding
canonical bases in
quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...
s. generalized Hall algebras to more general
categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
* Categories (Peirce)
* ...
, such as the category of representations of a
quiver
A quiver is a container for holding arrows, bolts, ammo, projectiles, darts, or javelins. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were trad ...
.
Construction
A
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
abelian ''p''-group ''M'' is a direct sum of
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
''p''-power components
where
is a
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of
called the ''type'' of ''M''. Let
be the number of subgroups ''N'' of ''M'' such that ''N'' has type
and the quotient ''M/N'' has type
. Hall proved that the functions ''g'' are
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
functions of ''p'' with integer coefficients. Thus we may replace ''p'' with an indeterminate ''q'', which results in the Hall polynomials
:
Hall next constructs an
associative ring
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying pro ...
over