|
Mutation (algebra)
In the theory of algebras over a field, mutation is a construction of a new binary operation related to the multiplication of the algebra. In specific cases the resulting algebra may be referred to as a homotope or an isotope of the original. Definitions Let ''A'' be an algebra over a field ''F'' with multiplication (not assumed to be associative) denoted by juxtaposition. For an element ''a'' of ''A'', define the left ''a''-homotope A(a) to be the algebra with multiplication :x * y = (xa)y. \, Similarly define the left (''a'',''b'') mutation A(a,b) :x * y = (xa)y - (yb)x. \, Right homotope and mutation are defined analogously. Since the right (''p'',''q'') mutation of ''A'' is the left (−''q'', −''p'') mutation of the opposite algebra to ''A'', it suffices to study left mutations.Elduque & Myung (1994) p. 34 If ''A'' is a unital algebra and ''a'' is invertible, we refer to the isotope by ''a''. Properties * If ''A'' is associative then so is any homotope o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebra Over A Field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set together with operations of multiplication and addition and scalar multiplication by elements of a field (mathematics), 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 where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed (but not excluded, either). Given an integer ''n'', the ring (mathematics), ring of real matrix, real square matrix, 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-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Binary Operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation on a set is a binary function that maps every pair of elements of the set to an element of the set. Examples include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement, intersection. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups. A binary function that involves several sets is sometimes also called a ''binary operation''. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Binary operations are the keystone of most structures that are studied in algebra, in parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isotopy Of An Algebra
In mathematics, an isotopy from a possibly non-associative algebra ''A'' to another is a triple of bijective linear maps such that if then . This is similar to the definition of an isotopy of loops, except that it must also preserve the linear structure of the algebra. For this is the same as an isomorphism. The autotopy group of an algebra is the group of all isotopies to itself (sometimes called autotopies), which contains the group of automorphisms as a subgroup. Isotopy of algebras was introduced by , who was inspired by work of Steenrod. Some authors use a slightly different definition that an isotopy is a triple of bijective linear maps ''a'', ''b'', ''c'' such that if then . For alternative division algebras such as the octonions the two definitions of isotopy are equivalent, but in general they are not. Examples *If is an isomorphism then the triple is an isotopy. Conversely, if the algebras have identity elements 1 that are preserved by the maps ''a'' and ''b'' of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Field (algebra)
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. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straightedge. Galois theory, devoted to understanding the symmetries of field extensions, provides an elegant proof of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Associative Property
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replacement for well-formed formula, expressions in Formal proof, logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the Operation (mathematics), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Opposite Algebra
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring whose multiplication ∗ is defined by for all in ''R''. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see '). Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc. Relation to automorphisms and antiautomorphisms In this section the symbol for multiplication in the opposite ring is changed from asterisk to diamond, to avoid confusing it with some unary operations. A ring is called a ''self-opposite'' ring if it is isomorphic to its opposite ring, which name indicates that R^\text is essentiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Unital Algebra
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 where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed (but not excluded, either). 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lie-admissible Algebra
In algebra, a Lie-admissible algebra, introduced by , is a (possibly non-associative) algebra that becomes a Lie algebra under the bracket 'a'', ''b''= ''ab'' − ''ba''. Examples include 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 mult ...s, Lie algebras, and Okubo algebras. See also * Malcev-admissible algebra * Jordan-admissible algebra References * * * * * * {{Authority control Non-associative algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Alternative Algebra
In abstract algebra, an alternative algebra is an algebra over a field, algebra in which multiplication need not be associative, only alternativity, alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating form, alternating. The associator is a trilinear map given by :[x,y,z] = (xy)z - x(yz). By definition, a multilinear map is alternating if it Vanish_(mathematics), vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to :[x,x,y] = 0 :[y,x,x] = 0 Both of these identities together imply that: :[x,y,x]=[x,x,x]+[x,y,x]+ :-[x,x+y,x+y] = := [x,x+y,-y] = := [x,x,-y] - [x,y,y] = 0 for all x and y. This is equivalent to the ''f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Malcev-admissible Algebra
In algebra, a Malcev-admissible algebra, introduced by , is a (possibly non-associative) algebra that becomes a Malcev algebra under the bracket 'a'', ''b''= ''ab'' − ''ba''. Examples include alternative algebras, Malcev algebras and Lie-admissible algebras. See also * Jordan-admissible algebra References * * * *{{citation , last=Myung , first=Hyo Chul , year=1986 , title=Malcev-admissible algebras , url=https://books.google.com/books?id=PBvvAAAAMAAJ , series= Progress in Mathematics , volume=64 , publisher=Birkhäuser Boston Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (partic ... , place=Boston, MA , isbn= 0-8176-3345-6 , mr=0885089 , doi=10.1007/978-1-4899-6661-2 Non-associative algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hurwitz Algebra
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bernstein Algebra
In mathematical genetics, a genetic algebra is a (possibly non-associative) algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras (also called weighted algebra). The study of these algebras was started by . In applications to genetics, these algebras often have a basis corresponding to the genetically different gametes, and the structure constants of the algebra encode the probabilities of producing offspring of various types. The laws of inheritance are then encoded as algebraic properties of the algebra. For surveys of genetic algebras see , and . Baric algebras Baric algebras (or weighted algebras) were introduced by . A baric algebra over a field ''K'' is a possibly non-associative algebra over ''K'' together with a homomorphism ''w'', called the weight, from the algebra to ''K''. Bern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
