In algebra, an elliptic algebra is a certain
regular algebra
In mathematics, a Kleene algebra ( ; named after Stephen Cole Kleene) is an idempotent (and thus partially ordered) semiring endowed with a closure operator. It generalizes the operations known from regular expressions.
Definition
Various ine ...
of a
Gelfand–Kirillov dimension In algebra, the Gelfand–Kirillov dimension (or GK dimension) of a right module ''M'' over a ''k''-algebra ''A'' is:
:\operatorname = \sup_ \limsup_ \log_n \dim_k M_0 V^n
where the supremum is taken over all finite-dimensional subspaces V \sub ...
three (
quantum polynomial ring In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry.
Examples
*The quantum plane, the most basic example, is the quotient ring of the free ring: ...
in three variables) that corresponds to a cubic divisor in the projective space P
2. If the cubic divisor happens to be an
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
, then the algebra is called a
Sklyanin algebra In mathematics, specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied in the classification of Artin-Schelter regular algebras of global dim ...
. The notion is studied in the context of
noncommutative projective geometry In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry.
Examples
*The quantum plane, the most basic example, is the quotient ring of the free ring: ...
.
References
*
{{algebra-stub
Algebraic structures
Algebraic logic