In some categories — notably groups, rings, and Lie algebras — it is possible to separate automorphisms into two types, called " inner " and " outer " automorphisms.

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

** the classical Lie groups, namely the simple groups related to the projective special linear, unitary, symplectic, or orthogonal transformations over a finite field ;

** the exceptional and twisted groups of Lie type ( including the Tits group ).

The simple groups of low 2-rank are mostly groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.

These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups.

This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.

As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic.

These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.

The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.

Watson wrote a series of didactic novels like Escaped from the Snare: Christian Science, Bewitched by Spiritualism, and The Gilded Lie, as warnings of the dangers posed by cultic groups.

Models that do not unify all interactions using one simple Lie group as the gauge symmetry, but do so using semisimple groups, can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well.

: For an elementary introduction to how Lie algebras are related to particle physics, see the article Particle physics and representation theory.

If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

* Lie groups — Many important Lie groups are compact, so the results of compact representation theory apply to them.

Other techniques specific to Lie groups are used as well.

Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.

* Linear algebraic groups ( or more generally affine group schemes ) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different ( and much less well understood ).

As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra and an ideal I in it, one constructs the factor algebra, and the first isomorphism theorem holds for Lie algebras.

Such Lie algebras are called abelian, cf.

In mathematics, a Lie group () is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

The complementary solvable Lie groups cannot in the same way be classified.

For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.

Such groups are classified using the prior classification of the complex simple Lie algebras: for which see the page on root systems.

The simple complex finite dimensional Lie superalgebras were classified by Victor Kac.

Similarly to how semisimple Lie algebras can be classified, the finite-dimensional representations of semisimple Lie algebras can be classified.

Semisimple Lie algebras are classified in terms of the weights of the adjoint representation, the so-called root system ; in a similar manner all finite-dimensional irreducible representations can be understood in terms of weights ; see weight ( representation theory ) for details.

This complements the earlier work of Lie himself, who had earlier classified the complex Lie algebras.

From 1888 to 1890, Killing essentially classified the complex finite dimensional simple Lie algebras, inventing the notions of a Cartan subalgebra and the Cartan matrix.

Each such example has a corresponding example of compact type, by considering a maximal compact subgroup of the complexification of G which contains K. More directly, the examples of compact type are classified by involutive automorphisms of compact simply connected simple Lie groups G ( up to conjugation ).

( He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.

Furthermore, for mathematicians, according to this formula the gluon color field can be represented by a SU ( 3 )- Lie algebra-valued " curvature "- 2-form where is a " vector potential "- 1-form corresponding to G and is the ( antisymmetric ) " wedge product " of this algebra, producing the " structure constants " f < sup > abc </ sup >.

However, according to Michael C. Moynihan ( Sept. 7, 2012 ) in Tablet Magazine, it has become common today in US politics today, to criticize an opponent for being guilty of using the idea of " technique " of the Big Lie.

In terms of Lie derivatives, according to the definition of the connection for the vector bundle.

Twelve Months, Eleven Days was released soon after the second single to little promotion and peaked at a disappointing number 34, which led to Sony recalling Barlow's ' greatest ' song according to critics, the scheduled third single from his second album " Lie to Me ".

Because every ( nonzero finite-dimensional ) representation of G has a one-dimensional invariant subspace according to the Lie – Kolchin theorem, every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one, which is another way to state the Lie – Kolchin theorem.

These developments provide a geometrical setting for algebraic methods ( using Lie sphere geometry ) and a classification of solutions according to 33 essentially different configurations of the given circles.

Suppose given a Dirac field ψ which transforms according to a ρ representation of the compact Lie group G ; and we have a background connection form of taking values in the Lie algebra The Dirac operator ( in Feynman slash notation ) is

The Lie bracket of two elements of L ( A ) is defined to be their commutator in A:

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings ; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.

These are not Lie groups because their underlying spaces are not real manifolds.

Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled

This can be used to reduce some problems about Lie groups ( such as finding their unitary representations ) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.

Isomorphic Lie groups are essentially the same ; they only differ in the notation for their elements.

Thus, manifolds, Lie groups, and graphs are all called connected if they are connected as topological spaces, and their components are the topological components.

* " The Story ", song by 30 Seconds to Mars from their album A Beautiful Lie

The relationship between the groups Sp ( 2n, C ), Sp ( 2n, R ) and Sp ( n ) is most evident at the level of their Lie algebras.

In the Zoroastrian faith, after death a soul must face judgment at the Bridge of the Separator ; there, evil people are tormented by prior denial of their own higher nature, or conscience, and " to all time will they be guests for the House of the Lie.

In that same chapter, Hitler accused " the Jews " of what he claimed was their use of the Big Lie.

They have released their eighteenth studio album C ' mon on July 27, 2011, which featured forty-ninth single " Don't Wanna Lie ".

Abelmann and Lie, ( 1997 ) report that the most profound result was the politicization of Korean Americans, all across the U. S. The younger generation especially realized they had been too uninvolved in American politics, and the riot shifted their political attention from South Korea to conditions in the United States.

Going to the mid-end of the ' 80s, Seattle had their thrash metal bands, such as The Accused ( a crossover thrash band ), Bitter End, Coven, Forced Entry, Terrorist a power trio featuring Andy Massey on guitar James Dean on bass and Vence LaRose drums formerly of Mace ... NME and R. I. P .. Also of particular note are Seattle's Slaughter Haus 5, Tacoma Bands Sword of Judgement, Hammer Head, Kil-D-Kor, Koz of Konfuzion, Diamond Lie featuring Jerry Cantrell of Alice in Chains, as well as Olympia bands Cyperus and Death Squad.

Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, and their representation theories.