In mathematics, the classical groups are defined as the special linear groups over the reals R {\displaystyle \mathbb {R} } , the complex numbers C {\displaystyle...

In Weyl's wonderful and terrible1 book The Classical Groups [W] one may discern two main themes: first, the study of the polynomial invariants for an arbitrary...

Artin investigated the orders of such groups, with a view to classifying cases of coincidence. A classical group is, roughly speaking, a special linear...

U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on C n {\displaystyle...

mathematics, the finite-dimensional representations of the complex classical Lie groups G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} , S L ( n , C )...

symplectic groups but it is possible to construct more using division algebras (for example the unit group of a quaternion algebra is a classical group). Note...

Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions...

foundational component of Western culture, classical music is frequently seen from the perspective of individual or groups of composers, whose compositions, personalities...

local fields, finite fields, and adele rings. The symplectic group is a classical group defined as the set of linear transformations of a 2n-dimensional...

isomorphic to the group of n {\displaystyle n} th roots of unity in the field F {\displaystyle F} . The so-called classical groups are subgroups of GL...

Classical liberalism is a political tradition and a branch of liberalism that advocates free market and laissez-faire economics and civil liberties under...

of view of Lie theory, the classical unitary group is a real form of the Steinberg group 2An, which is an algebraic group that arises from the combination...

of the classical groups and the groups of exceptional type in a 1955 paper. This omitted certain known groups (the projective unitary groups), which...

} ⁠. These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician...

Poincaré group, with rotations being produced as the composition of an even number of reflections. In classical physics, the Galilean group is a comparable...

Classics, also classical studies or Ancient Greek and Roman studies, is the study of classical antiquity. In the Western world, classics traditionally...

homogeneous space for the corresponding classical group. When k is strictly less than n then the special orthogonal group SO(n) also acts transitively on V...

Menace. With the music video for this theme, the LSO became the only classical group ever to have a video debut on MTV’s Total Request Live. "Duel of the...

^{3}} , which is equivalent to requiring them to preserve length. See classical group for a treatment of this more general approach, where SO(3) appears...

Indian classical dance, or Shastriya Nritya, is an umbrella term for different regionally-specific Indian classical dance traditions, rooted in predominantly...

mathematics, the circle group, denoted by T {\displaystyle \mathbb {T} } or ⁠ S 1 {\displaystyle \mathbb {S} ^{1}} ⁠, is the multiplicative group of all complex...

of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite groups often...

are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation,...

A classical planet is an astronomical object that is visible to the naked eye and moves across the sky and its backdrop of fixed stars (the common stars...

fundamental group. For compact Lie groups, there are two basic approaches to computing the fundamental group. The first approach applies to the classical compact...

rock band or the Baroque chamber group for basso continuo (harpsichord and cello) and one or more singers. In classical music, trios or quartets either...

space R {\displaystyle \mathbb {R} } n×n. Another classical group is the orthogonal group O(n), the group of all linear maps from R {\displaystyle \mathbb...

The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A n {\displaystyle A_{n}} , B n {\displaystyle...

The classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all...

dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period...