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Nilpotent Algebra (ring Theory)
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer ''n'' every product containing at least ''n'' elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a Lie algebra involves its Lie bracket, whereas, there is no Lie bracket defined in the general case of an algebra over a commutative ring.) Another possible source of confusion in terminology is the ''quantum nilpotent algebra'', a concept related to quantum groups and Hopf algebras. Formal definition An associative algebra A over a commutative ring R is defined to be a nilpotent algebra if and only if there exists some positive integer n such that 0=y_1\ y_2\ \cdots\ y_n for all y_1, \ y_2, \ \ldots,\ y_n in the algebra A. The smallest such n is called the index of the a ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity inste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Example Of A Non-associative Algebra
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if it is a vector space over ''K'' and is equipped with a ''K''- bilinear binary multiplication operation ''A'' × ''A'' → ''A'' which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (''ab'')(''cd''), (''a''(''bc''))''d'' and ''a''(''b''(''cd'')) may all yield different answers. While this use of ''non-associative'' means that associativity is not assumed, it does not mean that associativity is disallowed. In other words, "non-associative" means "not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nil-Coxeter Algebra
In mathematics, the nil-Coxeter algebra, introduced by , is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent. Definition The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by ''u''1, ''u''2, ''u''3, ... with the relations : \begin u_i^2 & = 0, \\ u_i u_j & = u_j u_i & & \text , i-j, > 1, \\ u_i u_j u_i & = u_j u_i u_j & & \text , i-j, =1. \end These are just the relations for the infinite braid group, together with the relations ''u'' = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations ''u'' = 0 to the relations of the corresponding generalized braid group. References *{{Citation , last1=Fomin , first1=Sergey , authorlink1=Sergey Fomin , last2=Stanley , first2=Richard P. , authorlink2=Richard P. Stanley , title=Schubert polynomials and the nil-Coxeter algebra , doi=10.1006/aima.1994.1009 , doi-acce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
