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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. [1][2] This class of groups contrasts with the abelian groups, where all pairs of group elements ...
Definition and first properties. The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that of function composition. [1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree is the ...
The Yang–Mills theory is a gauge theory based on a special unitary group SU (n), or more generally any compact Lie group. A Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U (1) × SU (2)) as ...
Statistical mechanics. In physics, an anyon is a type of quasiparticle so far observed only in two-dimensional systems. In three-dimensional systems, only two kinds of elementary particles are seen: fermions and bosons. Anyons have statistical properties intermediate between fermions and bosons. [1]
t. e. In physics, topological order[1] is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy [2] and quantized non-abelian geometric phases of degenerate ground states. [1] Microscopically, topological orders ...
The Standard Model predicts that each of these three numbers should be conserved separately in a manner similar to the way baryon number is conserved. These numbers are collectively known as lepton family numbers (LF). (This result depends on the assumption made in Standard Model that neutrinos are massless.
SU (2) is the universal covering group of SO (3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU (2) for the description of non-relativistic spin in theoretical physics; see below for other physical and historical context.
Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory ...