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Supercommutative Ring
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have :yx = (-1)^xy , where , ''x'', denotes the grade of the element and is 0 or 1 (in Z) according to whether the grade is even or odd, respectively. Equivalently, it is a superalgebra where the supercommutator : ,y= xy - (-1)^yx always vanishes. Algebraic structures which supercommute in the above sense are sometimes referred to as skew-commutative associative algebras to emphasize the anti-commutation, or, to emphasize the grading, graded-commutative or, if the supercommutativity is understood, simply commutative. Any commutative algebra is a supercommutative algebra if given the trivial gradation (i.e. all elements are even). Grassmann algebras (also known as exterior algebras) are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix ''super-'' comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. Formal definition Let ''K'' be a commutative ring. In most applications, ''K'' is a field of characteristic 0, such as R or C. A superalgebra over ''K'' is a ''K''-module ''A'' with a direct sum decomposition :A = A_0\oplus A_1 together with a bilinear multiplication ''A'' × ''A'' → ''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Algebra
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 R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Element
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 R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this articl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supercommutator
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ''even'' elements of the superalgebra correspond to bosons and ''odd'' elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around). Definition Formally, a Lie superalgebra is a nonassociative Z2- graded algebra, or ''superalgebra'', over a commutative ring (typically R or C) whose product ·, · called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading): Super skew-symmetry: : ,y-(-1)^ ,x\ The super Jacobi identity: :(-1)^ ,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^ ,_[y,_z_+_(-1)^[y,_[z,_x.html"_;"title=",_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1)^[y,_[z,_x">,_z.html"_;"title=",_[y,_z">,_[y,_z_+_(-1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even Subalgebra
In mathematics and theoretical physics, a superalgebra is a Z2- graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix ''super-'' comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. Formal definition Let ''K'' be a commutative ring. In most applications, ''K'' is a field of characteristic 0, such as R or C. A superalgebra over ''K'' is a ''K''-module ''A'' with a direct sum decomposition :A = A_0\oplus A_1 together with a bilinear multiplication ''A'' × ''A'' → ''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras. Examples *This definition can be applied in particular to square matrices. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See nilpotent matrix for more. * In the factor ring \Z/9\Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9. * Assume that two elements a and b in a ring R satisfy ab=0. Then the element c=ba is nilpotent as \beginc^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end An example with matrices (for ''a'', ''b''):A = \begin 0 & 1\\ 0 & 1 \end, \;\; B =\begin 0 & 1\\ 0 & 0 \end. Here AB=0 and BA=B. *By definition, any element of a nilsemigroup is nilpotent. Properties No nilpotent eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Algebra
In mathematics, an alternating algebra is a -graded algebra for which for all nonzero homogeneous elements and (i.e. it is an anticommutative algebra) and has the further property that for every homogeneous element of odd degree. Examples * The differential forms on a differentiable manifold form an alternating algebra. * The exterior algebra is an alternating algebra. * The cohomology ring of a topological space is an alternating algebra. Properties * The algebra formed as the direct sum of the homogeneous subspaces of even degree of an anticommutative algebra is a subalgebra contained in the centre of , and is thus commutative. * An anticommutative algebra over a (commutative) base ring in which 2 is not a zero divisor is alternating. See also * Alternating multilinear map * Exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplicatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded-commutative Ring
In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx, where , ''x'' , and , ''y'' , denote the degrees of ''x'' and ''y''. A commutative (non-graded) ring, with trivial grading, is a basic example. An exterior algebra is an example of a graded-commutative ring that is not commutative in the non-graded sense. A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative. In fact, many examples of graded-commutative rings come from algebraic topology and homological algebra. References * David Eisenbud, ''Commutative Algebra. With a view toward algebraic geometry'', Graduate Texts in Mathematics, vol 150, Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
