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Whitehead Product
In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in . The relevant MSC code is: 55Q15, Whitehead products and generalizations. Definition Given elements f \in \pi_k(X), g \in \pi_l(X), the Whitehead bracket : ,g\in \pi_(X) is defined as follows: The product S^k \times S^l can be obtained by attaching a (k+l)-cell to the wedge sum :S^k \vee S^l; the attaching map is a map :S^ \stackrel S^k \vee S^l. Represent f and g by maps :f\colon S^k \to X and :g\colon S^l \to X, then compose their wedge with the attaching map, as :S^ \stackrel S^k \vee S^l \stackrel X . The homotopy class of the resulting map does not depend on the choices of representatives, and thus one obtains a well-defined element of :\pi_(X). Grading Note that there is a shift of 1 in the grading (compared to the indexing of homotopy groups), so \pi_k(X) has degree (k-1); equivalently, L_k = \p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Lie Algebra
In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra. A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. These arise in the study of derivations on graded algebras, in the deformation theory of Murray Gerstenhaber, Kunihiko Kodaira, and Donald C. Spencer, and in the theory of Lie derivatives. A supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super \Z/2\Z-gradation. These arise when one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Lie Algebra
In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom : ,x0 replaced by : ,y- ,x/math> (anti-symmetry). In characteristic other than 2, these are equivalent (in the presence of bilinearity In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering ''Lie algebras'' over the integers. In a quasi-Lie algebra, :2 ,x0. Therefore, the bracket of any element with itself is 2-torsion, if it does not actually vanish. See also * Whitehead product References * Lie algebras {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or ''holes'', of a topological space. To define the ''n''-th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''-th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent ( homeomorphic), but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Subject Classification
The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020. Structure The MSC is a hierarchical scheme, with three levels of structure. A classification can be two, three or five digits long, depending on how many levels of the classification scheme are used. The first level is represented by a two-digit number, the second by a letter, and the third by another two-digit number. For example: * 53 is the classification for differential geometry * 53A is the classification for classical differential geometry * 53A45 is the classification for vector and tensor analysis First l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedge Sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the quotient space of the disjoint union of ''X'' and ''Y'' by the identification x_0 \sim y_0: X \vee Y = (X \amalg Y)\;/, where \,\sim\, is the equivalence closure of the relation \left\. More generally, suppose \left(X_i\right)_ is a indexed family of pointed spaces with basepoints \left(p_i\right)_. The wedge sum of the family is given by: \bigvee_ X_i = \coprod_ X_i\;/, where \,\sim\, is the equivalence closure of the relation \left\. In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints \left(p_i\right)_, unless the spaces \left(X_i\right)_ are homogeneous. The wedge sum is again a pointed space, and the binary operation is associative and commuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attaching Map
In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let ''X'' and ''Y'' be topological spaces, and let ''A'' be a subspace of ''Y''. Let ''f'' : ''A'' → ''X'' be a continuous map (called the attaching map). One forms the adjunction space ''X'' ∪''f'' ''Y'' (sometimes also written as ''X'' +''f'' ''Y'') by taking the disjoint union of ''X'' and ''Y'' and identifying ''a'' with ''f''(''a'') for all ''a'' in ''A''. Formally, :X\cup_f Y = (X\sqcup Y) / \sim where the equivalence relation ~ is generated by ''a'' ~ ''f''(''a'') for all ''a'' in ''A'', and the quotient is given the quotient topology. As a set, ''X'' ∪''f'' ''Y'' consists of the disjoint union of ''X'' and (''Y'' − ''A''). The topology, however, is specified by the quotient construction. Intuitively, one may think of ''Y'' as being glued onto ''X'' via the map ''f''. Examples *A common example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
