Irreducible representationIn mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.
Representation theory of finite groupsThe representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero.
Induced representationIn group theory, the induced representation is a representation of a group, G, which is constructed using a known representation of a subgroup H. Given a representation of H, the induced representation is, in a sense, the "most general" representation of G that extends the given one. Since it is often easier to find representations of the smaller group H than of G, the operation of forming induced representations is an important tool to construct new representations.
Lorentz transformationIn physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.
Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
SubgroupIn group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G).
Normal subgroupIn abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group.
Characteristic subgroupIn mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group. A subgroup H of a group G is called a characteristic subgroup if for every automorphism φ of G, one has φ(H) ≤ H; then write H char G.
Commutator subgroupIn mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, is abelian if and only if contains the commutator subgroup of . So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.
Adjoint representationIn mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: . For any Lie group, this natural representation is obtained by linearizing (i.
IP addressAn Internet Protocol address (IP address) is a numerical label such as 192.0.2.1 that is connected to a computer network that uses the Internet Protocol for communication. An IP address serves two main functions: network interface identification, and location addressing. Internet Protocol version 4 (IPv4) defines an IP address as a 32-bit number. However, because of the growth of the Internet and the depletion of available IPv4 addresses, a new version of IP (IPv6), using 128 bits for the IP address, was standardized in 1998.
Lorentz factorThe Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz. It is generally denoted γ (the Greek lowercase letter gamma).
Lorentz covarianceIn relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".
IP PBXAn IP PBX ("Internet Protocol private branch exchange") is a system that connects telephone extensions to the public switched telephone network (PSTN) and provides internal communication for a business. An IP PBX is a PBX system with IP connectivity and may provide additional audio, video, or instant messaging communication utilizing the TCP/IP protocol stack. Voice over IP (VoIP) gateways can be combined with traditional PBX functionality to allow businesses to use their managed intranet to help reduce long distance expenses and take advantage of the benefits of a single network for voice and data (converged network).
Voice over IPVoice over Internet Protocol (VoIP), also called IP telephony, is a method and group of technologies for voice calls, the delivery of voice communication sessions over Internet Protocol (IP) networks, such as the Internet. The broader terms Internet telephony, broadband telephony, and broadband phone service specifically refer to the provisioning of voice and other communications services (fax, SMS, voice messaging) over the Internet, rather than via the public switched telephone network (PSTN), also known as plain old telephone service (POTS).
History of Lorentz transformationsThe history of Lorentz transformations comprises the development of linear transformations forming the Lorentz group or Poincaré group preserving the Lorentz interval and the Minkowski inner product . In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group.
Group actionIn mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it.
Four-momentumIn special relativity, four-momentum (also called momentum–energy or momenergy) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum p = (px, py, pz) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is The quantity mv of above is ordinary non-relativistic momentum of the particle and m its rest mass.
Wigner's classificationIn mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since this group is noncompact, these unitary representations are infinite-dimensional.) It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states.
