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Graded vector space, the Glossary

In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers.^[1]

31 relations: Abelian group, Associative algebra, Cancellation property, Category (mathematics), Comodule, Degree of a polynomial, Direct sum of modules, Embedding, Endomorphism ring, Field (mathematics), Formal power series, Graded ring, Graded structure, Hilbert–Poincaré series, Homological algebra, Integer, Linear map, Linear subspace, Littlewood–Richardson rule, Mathematics, Monoid, Monomial, Morphism, Natural number, Nicolas Bourbaki, Physics, Polynomial, Ring (mathematics), Semigroup, Super vector space, Vector space.
Categories in category theory
Vector spaces

Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

See Graded vector space and Abelian group

Associative algebra

In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).

See Graded vector space and Associative algebra

Cancellation property

In mathematics, the notion of cancellativity (or cancellability) is a generalization of the notion of invertibility.

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Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows".

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Comodule

In mathematics, a comodule or corepresentation is a concept dual to a module.

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Degree of a polynomial

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients.

See Graded vector space and Degree of a polynomial

Direct sum of modules

In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.

See Graded vector space and Direct sum of modules

Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

See Graded vector space and Embedding

Endomorphism ring

In mathematics, the endomorphisms of an abelian group X form a ring.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.

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Formal power series

In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form where the a_n, called coefficients, are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable.

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Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that.

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Graded structure

In mathematics, the term "graded" has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts.

See Graded vector space and Graded structure

Hilbert–Poincaré series

In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite).

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Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.

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Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

See Graded vector space and Integer

Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.

See Graded vector space and Linear map

Linear subspace

In mathematics, and more specifically in linear algebra, a linear subspace or vector subspaceThe term linear subspace is sometimes used for referring to flats and affine subspaces.

See Graded vector space and Linear subspace

Littlewood–Richardson rule

In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Monoid

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.

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Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.

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Morphism

In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces.

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Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.

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Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS).

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Physics

Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

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Polynomial

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

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Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

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Semigroup

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.

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Super vector space

In mathematics, a super vector space is a \mathbb Z_2-graded vector space, that is, a vector space over a field \mathbb K with a given decomposition of subspaces of grade 0 and grade 1. Graded vector space and super vector space are Categories in category theory.

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Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''. Graded vector space and vector space are vector spaces.

See Graded vector space and Vector space

References

[1] https://en.wikipedia.org/wiki/Graded_vector_space

Also known as Category of graded vector spaces, Direct sum of graded vector spaces, Graded dimension, Graded linear map, Graded linear space, Tensor product of graded vector spaces.