* If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application defined by is continuous.

* The kernel of a representation ρ of a group G is defined as the normal subgroup of G whose image under ρ is the identity transformation:

A monomial representation of a group G is a linear representation ρ: G → GL ( n, F ) of G ( here F is the defining field of the representation ) such that the image ρ ( G ) is a subgroup of the group of monomial matrices.

Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ ( g ) is a unitary operator for all g ∈ G ; i. e., for all v, w ∈ H. ( I. e.

* Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.

To state the third and final part of the theorem, there is a natural Hilbert space over G consisting of square-integrable functions, L < sup > 2 </ sup >( G ); this makes sense because Haar measure exists on G. Calling this Hilbert space H, the group G has a unitary representation ρ on H by acting on the left, via

Given a state ρ of A, there is a *- representation π of A with distinguished cyclic vector ξ such that its associated state is ρ, i. e.

The vector space V, together with the representation ρ, is called a-module.

If L is a real Lie algebra and ρ: L × V → V is a complex representation of it, we can construct another representation of L called its dual representation as follows.

One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation.

For the right regular representation ρ, an inversion must occur in order to satisfy the axioms of a representation.

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra, into which the spin group Spin ( p, q ) may be embedded.

In this construction the representation of the Clifford algebra Cℓ ( V, g ), the Lie algebra so ( V, g ), and the Spin group Spin ( V, g ), all depend on the choice of the orthonormal basis and the choice of the gamma matrices.

* is the real 6D vector space corresponding to the vector representation of Spin ( 4, 2 ).

Spin networks provide a representation for computations of amplitudes between two different hypersurfaces of the manifold.

This automorphism group permutes the three 8-dimensional irreducible representations of Spin ( 8 ); these being the vector representation and two chiral spin representations.

The vector representation – the natural action of SO ( 8 ) ( hence Spin ( 8 )) on – is also known as the " defining module ", while the chiral spin representations are also known as " half-spin representations ", and all three of these are fundamental representations.

No other Dynkin diagram has an automorphism group of order greater than 2 ; for other D < sub > n </ sub > ( corresponding to other even Spin groups, Spin ( 2n )), there is still the automorphism corresponding to switching the two half-spin representations, but these are not isomorphic to the vector representation.

By restricting this to the unit quaternions, we obtain a quaternionic representation of the spinor group Spin ( 3 ).

Another unitary example is the spin representation of Spin ( 5 ).

An example of a nonunitary quaternionic representation would be the two dimensional irreducible representation of Spin ( 5, 1 ).

The project involved 3, 400 workers, mostly immigrants from Europe, along with hundreds of Mohawk iron workers, many from the Kahnawake reserve near Montreal. Perhaps the most famous popular culture representation of the building is in the 1933 film King Kong, in which the title character, a giant ape, climbs to the top to escape his captors but falls to his death after being attacked by airplanes.

Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers ( such as 2, 3, 4, 6, 8, 9 ...) have a terminating representation in duodecimal.

In particular, the five most elementary fractions ( ½, ⅓, ⅔, ¼ and ¾ ) all have a short terminating representation in duodecimal ( 0. 6, 0. 4, 0. 8, 0. 3 and 0. 9, respectively ), and twelve is the smallest radix with this feature ( since it is the least common multiple of 3 and 4 ).

The concentration of water associated with iron oxide varies, thus chemical representation is presented as Fe < sub > 2 </ sub > O < sub > 3 </ sub >· xH < sub > 2 </ sub > O.

The oldest known iconographic representation of an instrument displaying the essential features of a guitar is a 3, 300-year-old stone carving of a Hittite bard.

* ISO 3166-3, Codes for the representation of names of countries and their subdivisions – Part 3: Code for formerly used names of countries, defines codes for country names which have been deleted from ISO 3166-1 since its first publication in 1974.

Pi Day is observed on March 14 because of the date's representation as 3 / 14 in month / day date format.

This representation adheres to the commonly used approximation of 3. 14 for π.

where is the quark field, a dynamical function of space-time, in the fundamental representation of the SU ( 3 ) gauge group, indexed by ; are the gluon fields, also a dynamical function of space-time, in the adjoint representation of the SU ( 3 ) gauge group, indexed by a, b, ....

Quarks are represented by Dirac fields in the fundamental representation 3 of the gauge group SU ( 3 ).

Gluons are spin-1 bosons which also carry color charges, since they lie in the adjoint representation 8 of SU ( 3 ).

In 1924 a new constitution was put in place, introducing elected representation ( 3 out of 22 members ) for the first time.

The term wireframe comes from designers using metal wire to represent the 3 dimensional shape of solid objects. 3D wireframe allows to construct and manipulate solids and solid surfaces. 3D solid modeling technique efficiently draws high quality representation of solids than the conventional line drawing.

On 5 March 2004, Dr. Mitchell unveiled a 3 / 4 scale representation of the prototype Spitfire K5054 at the entrance to Eastleigh Airport, Southampton, on the 68th anniversary of its first flight.

At the 2005 parliamentary elections, the SDPL gained only 3. 9 % of the vote, which was insufficient for the alliance to achieve parliamentary representation.

Thirteen of Great Britain's colonies rebelled with the American Revolutionary War, beginning in 1775, primarily over representation, local laws and tax issues, and established the United States of America, which was recognized internationally with the signing of the Treaty of Paris ( 1783 ) on September 3 of that year ( 1783 ).

: 1. 125 2 < sup > 3 </ sup > ( using decimal representation ), or, in E notation,

The elections take place within a system of proportional representation ( Article 129 sub 2 ); the vote is secret and the organisation of the voting is regulated by law ( Subarticle 3 ).

An alternative representation is: " 40: 60 :: 2: 3 "

Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

The ( unique ) representable functor F: → is the Cayley representation of G. In fact, this functor is isomorphic to and so sends to the set which is by definition the " set " G and the morphism g of ( i. e. the element g of G ) to the permutation F < sub > g </ sub > of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group

: A faithful representation is one in which the homomorphism G → GL ( V ) is injective ; in other words, one whose kernel is the trivial subgroup

In fact another theoretical representation of an amine oxide uses a dative covalent bond ≡ N → O with no formal charges.

In fancy language, this amounts to saying that a limit of F is a representation of the functor Cone (–, F ): C → Set.

But every faithful G-set ( equivalently, every representation of G as a group of permutations ) determines a faithful functor G → Set.

A typical interaction vertex ( called the three gluon vertex ) for gluons involves g + g → g. This is shown here, along with its color-line representation.

Let V be a representation of a Lie algebra g over a field F and let λ be a weight of g. Then the weight space of V with weight λ: g → F is the subspace

A representation of a Lie group G on a finite-dimensional complex vector space V is a smooth group homomorphism Ψ: G → Aut ( V ) from G to the automorphism group of V.

A representation of a Lie group G on a vector space V ( over a field K ) is a smooth ( i. e. respecting the differential structure ) group homomorphism G → Aut ( V ) from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into GL ( n, K ).

A representation of a Lie group G on a complex Hilbert space V is a group homomorphism Ψ: G → B ( V ) from G to B ( V ), the group of bounded linear operators of V which have a bounded inverse, such that the map G × V → V given by ( g, v ) → Ψ ( g ) v is continuous.

We define the representation Ψ: R → B ( L < sup > 2 </ sup >( R )) by Ψ ( r )

where π: G → GL ( V ) is a finite-dimensional ( continuous ) group representation of G, and L is a linear functional on the vector space of endomorphisms of V ( e. g. trace ), which contains GL ( V ) as an open subset.

However, one can lift a projective representation of G to a linear representation of a different group C, which will be a central extension of G. To understand this, note that GL ( V ) → PGL ( V ) is a central extension of PGL, meaning that the kernel is central ( in fact, is exactly the center of GL ).

As a consequence, the kernel of the Gelfand representation A → C < sub > 0 </ sub >( Φ < sub > A </ sub >) may be identified with the Jacobson radical of A.