* In representation theory, Lusztig's canonical basis and closely related Kashiwara's crystal basis in quantum groups and their representations

Langlands proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible representations.

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.

All irreducible unitary representations are admissible ( or rather their Harish-Chandra modules are ), and the admissible representations are given by the Langlands classification, and it is easy to tell which of them have a non-trivial invariant sesquilinear form.

is a classification of the nonnegative () energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues.

The search for examples ultimately led to a complete classification of irreducible affine holonomies by Merkulov and Schwachhöfer ( 1999 ), with Bryant ( 2000 ) showing that every group on their list occurs as an affine holonomy group.

The classification of irreducible real affine holonomies can be obtained from a careful analysis, using the lists above and the fact that real affine holonomies complexify to complex ones.

* 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 ).

At the other end of the scale, there were representations from countries and areas where a detailed and sophisticated classification was irrelevant, but which nevertheless needed a classification based on the ICD in order to assess their progress in health care and in the control of disease.

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 language of Clifford algebras ( also called geometric algebras ) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras.

For example, over the real numbers, this is the Langlands classification of representations of real reductive groups.

The same classification applies to discrete subgroups of, the binary polyhedral groups ; properly, binary polyhedral groups correspond to the simply laced affine Dynkin diagrams and the representations of these groups can be understood in terms of these diagrams.

This idea led naturally to an analysis of the representation theory of semi-direct products in terms of ergodic actions of groups and in some cases a complete classification of such representations.

For a discussion of such unitary representations, see Wigner's classification.

Part 1, deals with the classification of crystalline substances by space groups and is not a numerical data compilation.

Other numerically important groups include the closely interrelated Chokwe and Lunda, the Ganguela and Nhaneca-Humbe, in both cases classification terms which stand for a variety of small groups, the Ovambo, the Herero, the Xindonga and scattered residual groups of Khoisan.

Based on phylogenetic research, the latest ( 2009 ) revision of the APG classification groups together the former families Hemerocallidaceae, Xanthorrhoeaceae sensu stricto and Asphodelaceae as the Xanthorrhoeaceae.

Based on phylogenetic research, the latest ( 2009 ) revision of the APG classification groups together these three families under the conserved name of Amaryllidaceae.

The subfamilial and tribal classification for the family is currently in a state of flux, with many of the groups being found to be grossly paraphyletic / polyphyletic.

Another classification system is based on biological activity ; in this classification, antibacterials are divided into two broad groups according to their biological effect on microorganisms: bactericidal agents kill bacteria, and bacteriostatic agents slow down or stall bacterial growth.

No classification of patients in subtypes and groups of subtypes is adequate.

The term black people is used in some socially-based systems of racial classification for humans of a dark-skinned phenotype, relative to other racial groups represented in a particular social context.

The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups.

These groups are sometimes classified under denominations, though for theological reasons many groups reject this classification system.

With the publication of Darwin's theory of evolution in 1859, the concept of a " natural system " of taxonomy gained a theoretical basis, and the idea was born that groups used in a system of classification should represent branches on the evolutionary tree of life.

They proposed the classification of legal system into seven groups, or so-called ' families ', in particular the

David proposed the classification of legal systems, according to the different ideology inspiring each one, into five groups or families:

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below.

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups ( and their action on other mathematical objects ) can sometimes be reduced to questions about finite simple groups.

Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups.

The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.

Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often ( but not always ) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods.

Lie, Hwang, and Tillman developed a complete survey along with a systematic classification of availability.

Later Jordan's results on classical groups were generalized to arbitrary finite fields by Leonard Dickson, following the classification of complex simple Lie algebras by Wilhelm Killing.

They also appear prominently in the classification of Lie groups.

In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp ( 2n, C ) is denoted C < sub > n </ sub >, and Sp ( n ) is the compact real form of Sp ( 2n, C ).

Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory ( such as singularity theory ).

( This list is closely related to, but not identical with, the Bianchi classification of the three-dimensional real Lie algebras into nine classes.

It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry.

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

These exceptional groups were discovered around 1890 in the classification of the simple Lie algebras, over the complex numbers ( Wilhelm Killing, re-done by Élie Cartan ).

The designation E < sub > 6 </ sub > comes from the Cartan – Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub >, D < sub > n </ sub >, and five exceptional cases labeled E < sub > 6 </ sub >, E < sub > 7 </ sub >, E < sub > 8 </ sub >, F < sub > 4 </ sub >, and G < sub > 2 </ sub >.

Moreover a supergroup has a super Lie algebra which plays a role similar to that of a Lie algebra for Lie groups in that they determine most of the representation theory and which is the starting point for classification.

Schneider and N. Andruskiewitsch finished their long-term classification effort of pointed Hopf algebras with coradical an abelian group ( excluding primes 2, 3, 5, 7 ), especially as the above finite quotients of Just like ordinary Semisimple Lie algebra they decompose into E ´ s ( Borel part ), dual F ´ s and K ´ s ( Cartan algebra ):

On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not locally symmetric: one of them, known as Berger's first criterion, is a consequence of the Ambrose – Singer theorem, that the curvature generates the holonomy algebra ; the other, known as Berger's second criterion, comes from the requirement that the connection should not be locally symmetric.