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Elliptic Algebra
In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three ( quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective space P2. 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. I ..., then the algebra is called a Sklyanin algebra. The notion is studied in the context of noncommutative projective geometry. References * {{algebra-stub Algebraic structures Algebraic logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 inequivalent definitions of Kleene algebras and related structures have been given in the literature. Here we will give the definition that seems to be the most common nowadays. A Kleene algebra is a set ''A'' together with two binary operations + : ''A'' × ''A'' → ''A'' and · : ''A'' × ''A'' → ''A'' and one function * : ''A'' → ''A'', written as ''a'' + ''b'', ''ab'' and ''a''* respectively, so that the following axioms are satisfied. * Associativity of + and ·: ''a'' + (''b'' + ''c'') = (''a'' + ''b'') + ''c'' and ''a''(''bc'') = (''ab'')''c'' for all ''a'', ''b'', ''c'' in ''A''. * Commutativity of +: ''a'' + ''b'' = ''b'' + ''a'' for all ''a'', ''b'' in ''A'' * Distributivity: ''a''(''b'' + ''c'') = (''ab'') + (''ac' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 \subset A and M_0 \subset M. An algebra is said to have polynomial growth if its Gelfand–Kirillov dimension is finite. Basic facts *The Gelfand–Kirillov dimension of a finitely generated commutative algebra ''A'' over a field is the Krull dimension of ''A'' (or equivalently the transcendence degree of the field of fractions of ''A'' over the base field.) *In particular, the GK dimension of the polynomial ring k _1, \dots, x_n/math> Is ''n''. *(Warfield) For any real number ''r'' ≥ 2, there exists a finitely generated algebra whose GK dimension is ''r''. In the theory of D-Modules Given a right module ''M'' over the Weyl algebra A_n, the Gelfand–Kirillov dimension of ''M'' over the Weyl algebra coincides with the dimension of ''M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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: ::k \langle x, y \rangle / (yx - q xy) *More generally, the quantum polynomial ring is the quotient ring: ::k \langle x_1, \dots, x_n \rangle / (x_i x_j - q_ x_j x_i) Proj construction By definition, the Proj of a graded ring ''R'' is the quotient category of the category of finitely generated graded modules over ''R'' by the subcategory of torsion modules. If ''R'' is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of ''R'' in this sense is equivalent to the category of coherent sheaves on the usual Proj of ''R''. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring. See also *Elliptic algebra In algebra, an elliptic algebra is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ... in .) It is always understood that the curve is really sitting in the projective plane, with the point being the uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 dimension 3 in the 1980s. Sklyanin algebras can be grouped into two different types, the non-degenerate Sklyanin algebras and the degenerate Sklyanin algebras, which have very different properties. A need to understand the non-degenerate Sklyanin algebras better has led to the development of the study of point modules in noncommutative geometry. Formal definition Let k be a field with a primitive cube root of unity. Let \mathfrak be the following subset of the projective plane \textbf_k^2: \mathfrak = \ \sqcup \. Each point :b:c\in \textbf_k^2 gives rise to a (quadratic 3-dimensional) Sklyanin algebra, S_ = k \langle x,y,z \rangle / (f_1, f_2, f_3), where, f_1 = ayz + bzy + cx^2, \quad f_2 = azx + bxz + cy^2, \quad f_3 = axy + b yx + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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: ::k \langle x, y \rangle / (yx - q xy) *More generally, the quantum polynomial ring is the quotient ring: ::k \langle x_1, \dots, x_n \rangle / (x_i x_j - q_ x_j x_i) Proj construction By definition, the Proj of a graded ring ''R'' is the quotient category of the category of finitely generated graded modules over ''R'' by the subcategory of torsion modules. If ''R'' is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of ''R'' in this sense is equivalent to the category of coherent sheaves on the usual Proj of ''R''. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring. See also *Elliptic algebra * Calabi–Yau algebra *Sklyanin algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Structures
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called ''scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
