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.

* 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 ).

* Mathematics: Approximation theory — Arakelov theory — Asymptotic theory — Bifurcation theory — Catastrophe theory — Category theory — Chaos theory — Choquet theory — Coding theory — Combinatorial game theory — Computability theory — Computational complexity theory — Deformation theory — Dimension theory — Ergodic theory — Field theory — Galois theory — Game theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory — Intersection theory — Invariant theory — Iwasawa theory — K-theory — KK-theory — Knot theory — L-theory — Lie theory — Littlewood – Paley theory — Matrix theory — Measure theory — Model theory — Morse theory — Nevanlinna theory — Number theory — Obstruction theory — Operator theory — PCF theory — Perturbation theory — Potential theory — Probability theory — Ramsey theory — Rational choice theory — Representation theory — Ring theory — Set theory — Shape theory — Small cancellation theory — Spectral theory — Stability theory — Stable theory — Sturm – Liouville theory — Twistor theory

Many gun shops have jointly participated in programs ( such as: “ Don ’ t Lie For The Other Guy ”) to deter such purchases.

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.

* An important class of infinite-dimensional real Lie algebras arises in differential topology.

Linear actions of Lie groups are especially important, and are studied in representation theory.

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.

This is important, because it allows generalization of the notion of a Lie group to Lie supergroups.

Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries.

The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics.

Engel would help Lie to write his most important treatise, Theorie der Transformationsgruppen, published in Leipzig in three volumes from 1888 to 1893.

These groups provide important examples of Lie groups.

Two important special cases of this are the exponential map for a manifold with a Riemannian metric, and the exponential map from a Lie algebra to a Lie group.

The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.

Since Lie groups ( and some analogues such as algebraic groups ) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied.

In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry.

One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real reductive Lie groups.

Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra.

The Lie derivative also has important properties when acting on differential forms.

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.

Other techniques specific to Lie groups are used as well.

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.

A GUT model basically consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang-Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra ( which is specified by a coupling constant for each factor ), a Higgs sector consisting of a number of scalar fields taking on values within real / complex representations of the Lie group and chiral Weyl fermions taking on values within a complex rep of the Lie group.

The semisimple Lie groups have a deep theory, building on the compact case.

It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle.

* The unitary group U ( n ) consisting of n × n unitary matrices ( with complex entries ) is a compact connected Lie group of dimension n < sup > 2 </ sup >.

This is a one-dimensional compact connected abelian Lie group.

The work of Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights.

The Lie algebra of any compact Lie group ( very roughly: one for which the symmetries form a bounded set ) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones.

Hence the question arises: what are the simple Lie algebras of compact groups?

* Compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S < sup > 1 </ sup > and simple compact Lie groups ( which correspond to connected Dynkin diagrams ).

The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected ( though it does map onto the Lie group for connected groups that are either compact or nilpotent ).