Nonabelian Homological Algebra

	Nonabelian Homological Algebra Non-abelian or nonabelian may refer to: * Non-abelian group, in mathematics, a group that is not abelian (commutative) * Non-abelian gauge theory, in physics, a gauge group that is non-abelian See also * Non-abelian gauge transformation, a gauge transformation * Non-abelian class field theory, in class field theory * Nonabelian cohomology, a cohomology * Abelian (other) Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ... * {{Mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Non-abelian Group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ''b'' ≠ ''b'' ∗ ''a''. This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute). Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Non-abelian Gauge Theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called ''gauge bosons' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Non-abelian Gauge Transformation In theoretical physics, a non-abelian gauge transformation means a gauge transformation taking values in some group ''G'', the elements of which do not obey the commutative law when they are multiplied. By contrast, the original choice of gauge group in the physics of electromagnetism had been U(1), which is commutative. For a non-abelian Lie group ''G'', its elements do not commute, i.e. they in general do ''not'' satisfy :ab=ba \,. The quaternions marked the introduction of non-abelian structures in mathematics. In particular, its generators t^a, which form a basis for the vector space of infinitesimal transformations (the Lie algebra), have a commutation rule: :\left ^a,t^b\right= t^a t^b - t^b t^a = C^ t_c. The structure constants C^ quantify the lack of commutativity, and do not vanish. We can deduce that the structure constants are antisymmetric in the first two indices and real. The normalization is usually chosen (using the Kronecker delta) as :Tr(t^at^b) = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Non-abelian Class Field Theory In mathematics, non-abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field ''K'', to the general Galois extension ''L''/''K''. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense. History A presentation of class field theory in terms of group cohomology was carried out by Claude Chevalley, Emil Artin and others, mainly in the 1940s. This resulted in a formulation of the central results by means of the group cohomology of the idele class group. The theorems of the cohomological approach are independent of whether or not the Galois group ''G'' of ''L''/''K'' is abelian. This theory has never been regarded as the sought-after ''non-abelian'' theory. The first reason that can be cited for that is that it did not provide fresh information on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Nonabelian Cohomology In mathematics, a nonabelian cohomology is any cohomology with coefficients in a nonabelian group, a sheaf of nonabelian groups or even in a topological space. If homology is thought of as the abelianization of homotopy (cf. Hurewicz theorem), then the nonabelian cohomology may be thought of as a dual of homotopy groups. Nonabelian Poincaré duality SeeNonabelian Poincare Duality (Lecture 8) See also * Stacks Group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ... References * {{topology-stub Cohomology theories ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]