* Milne, J. S., Affine Group Schemes ; Lie Algebras ; Lie Groups ; Reductive Groups ; Arithmetic Subgroups

Other types of explanation are Deductive-nomological, Functional, Historical, Psychological, Reductive, Teleological, Methodological explanations.

Reductive work-up conditions are far more commonly used than oxidative conditions.

Reductive physicalism, a form of monism, is incompatible with panpsychism.

Reductive dechlorination is a term that is used to describe certain types of degradation of chlorinated organic compounds by chemical reduction with release of inorganic chloride ions.

* ( 2001 ) " Conceptual Analysis and Reductive Explanation ", Philosophical Review, 110, 3, pp. 315-360 ( with David J. Chalmers ).

Reductive materialism being at one end of a continuum ( our theories will reduce to facts ) and eliminative materialism on the other ( certain theories will need to be eliminated in light of new facts ), Revisionary materialism is somewhere in the middle.

Reductive amination ( also known as reductive alkylation ) is a form of amination that involves the conversion of a carbonyl group to an amine via an intermediate imine.

* Reductive method: In the triple bottom line, a corporate-oriented approach, the social — that is, the way in which humans live and relate to each other and the environment — is secondary.

Reductive group schemes-develops the theory from the scheme-theoretic viewpoint.

Reductive elimination of the desired product 9 restores the original palladium catalyst 1.

* Reductive elimination gives rise to the desired coupling product.

Reductive iron acquisition includes conversion of iron from the ferric ( Fe < sup >+ 3 </ sup >) to the ferrous ( Fe < sup >+ 2 </ sup >) state and subsequent uptake via FtrA, an iron permease.

Reductive elimination is the reverse of oxidative addition.

Reductive elimination is favored when the newly formed X – Y bond is strong.

Reductive elimination is the key product-releasing step of several reactions that form C – H and C – C bonds.

Reductive amination reactions such as this one will not produce quaternary ammonium salts, but instead will stop at the tertiary amine stage.

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

For Lie algebras the definition is slightly different.

Lie algebras: Assigning to every real ( complex ) Lie group its real ( complex ) Lie algebra defines a functor.

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

Note: These models refer to Lie algebras not to Lie groups.

See Representations of Lie groups and Representations of Lie algebras.

* 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 ).

Lie algebras were introduced to study the concept of infinitesimal transformations.

Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras.

A homomorphism between two Lie algebras ( over the same ground field ) is a linear map that is compatible with the commutators:

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.

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

Such Lie algebras are called abelian, cf.

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

A new method for classifying complex filiform Lie algebras, Applied Mathematics and Computation, 121 ( 2-3 ): 169 – 175, 2001

* Jacobson, Nathan, Lie algebras, Republication of the 1962 original.

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.

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.

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.

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.