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Valya Algebra
In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra ''M'' over a field ''F'' whose multiplicative binary operation ''g'' satisfies the following axioms: 1. The skew-symmetry condition :g (A, B) =-g (B, A) for all A,B \in M. 2. The Valya identity : J (g (A_1, A_2), g (A_3, A_4), g (A_5, A_6)) =0 for all A_k \in M, where k=1,2,...,6, and J (A, B, C):= g (g (A, B), C)+g (g (B, C), A)+g (g (C, A), B). 3. The bilinear condition : g(aA+bB,C)=ag(A,C)+bg(B,C) for all A,B,C \in M and a,b \in F. We say that M is a Valya algebra if the commutant of this algebra is a Lie subalgebra. Each Lie algebra is a Valya algebra. There is the following relationship between the commutant-associative algebra and Valentina algebra. The replacement of the multiplication g(A,B) in an algebra M by the operation of commutation ,Bg(A,B)-g(B,A), makes it into the algebra M^. If M is a commutant-associative algebra, then M^ is a Valya algebra. A Valya alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anatoly Maltsev
Anatoly Ivanovich Maltsev (also: Malcev, Mal'cev; Russian: Анато́лий Ива́нович Ма́льцев; 27 November N.S./14 November O.S. 1909, Moscow Governorate – 7 June 1967, Novosibirsk) was born in Misheronsky, near Moscow, and died in Novosibirsk, USSR. He was a mathematician noted for his work on the decidability of various algebraic groups. Malcev algebras (generalisations of Lie algebras), as well as Malcev Lie algebras are named after him. Biography At school, Maltsev demonstrated an aptitude for mathematics, and when he left school in 1927, he went to Moscow State University to study Mathematics. While he was there, he started teaching in a secondary school in Moscow. After graduating in 1931, he continued his teaching career and in 1932 was appointed as an assistant at the Ivanovo Pedagogical Institute located in Ivanovo, near Moscow. Whilst teaching at Ivanovo, Maltsev made frequent trips to Moscow to discuss his research with Kolmogorov. Maltsev's f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Gennadievich Kurosh
Aleksandr Gennadyevich Kurosh (russian: Алекса́ндр Генна́диевич Ку́рош; January 19, 1908 – May 18, 1971) was a Soviet mathematician, known for his work in abstract algebra. He is credited with writing ''The Theory of Groups'', the first modern and high-level text on group theory, published in 1944. He was born in Yartsevo, in the Dukhovshchinsky Uyezd of the Smolensk Governorate of the Russian Empire and died in Moscow. He received his doctorate from the Moscow State University in 1936 under the direction of Pavel Alexandrov. In 1937 he became a professor there, and from 1949 until his death he held the Chair of Higher Algebra at Moscow State University. In 1938, he was the PhD thesis adviser to his fellow group theory scholar Sergei Chernikov, with whom he would develop important relationships between finite and infinite groups, discover the Kurosh-Chernikov class of groups, and publish several influential papers over the next decades. In al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutant-associative Algebra
In abstract algebra, a commutant-associative algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom: : ( _1,A_2 _3,A_4 _5,A_6 =0 , where 'A'', ''B''nbsp;= ''AB'' − ''BA'' is the commutator of ''A'' and ''B'' and (''A'', ''B'', ''C'') = (''AB'')''C'' – ''A''(''BC'') is the associator of ''A'', ''B'' and ''C''. In other words, an algebra ''M'' is commutant-associative if the commutant, i.e. the subalgebra of ''M'' generated by all commutators 'A'', ''B'' is an associative algebra. See also * Valya algebra * Malcev algebra * Alternative algebra References * A. Elduque, H. C. Myung ''Mutations of alternative algebras'', Kluwer Academic Publishers, Boston, 1994, * * M.V. Karasev, V.P. Maslov, ''Nonlinear Poisson Brackets: Geometry and Quantization.'' American Mathematical Society, Providence, 1993. * A.G. Kurosh, ''Lectures on general algebra.'' Translated from the Russian e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Algebra
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only 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. The associator is a trilinear map given by : ,y,z= (xy)z - x(yz). By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent toSchafer (1995) p. 27 : ,x,y= 0 : ,x,x= 0. Both of these identities together imply that : ,y,x= , x, x+ , y, x- , x+y, x+y= , x+y, -y= , x, -y- , y, y= 0 for all x and y. This is equivalent to the ''flexible identity''Schafer (1995) p. 28 :(xy)x = x(yx). The associator of an al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Malcev Algebra
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that :xy = -yx and satisfies the Malcev identity :(xy)(xz) = ((xy)z)x + ((yz)x)x + ((zx)x)y. They were first defined by Anatoly Maltsev (1955). Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop. Examples *Any Lie algebra is a Malcev algebra. *Any alternative algebra may be made int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Field
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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have an identity element. A quasigroup with an identity element is called a loop. Definitions There are at least two structurally equivalent formal definitions of quasigroup. One defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup defined with a single binary operation, however, need not be a quasigroup. We begin with the first definition. Algebra A quasigroup is a non-empty set ''Q'' with a binary operation ∗ (that is, a magma, indicating that a quasigroup has to satisfy closure property), obeying the Latin square property. This states that, for each ''a'' and ''b'' in ''Q'', there exist uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
