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.

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.

That is is the Lie derivative of f in the direction X, and one has df ( X )= X ( f ).

Then is a derivation and is linear, i. e., and, and a Lie algebra homomorphism, i. e.,, but it is not always an algebra homomorphism, i. e. the identity does not hold in general.

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.

If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in " standard form " meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.

The problem is to show that if a group has a centralizer of involution in " standard form " then it is a group of Lie type of odd characteristic.

* 1946 – Trygve Lie of Norway is picked to be the first United Nations Secretary General.

The smallest simple Lie group which contains the standard model, and upon which the first Grand Unified Theory was based, is

The next simple Lie group which contains the standard model is

This is a significant result, as other Lie groups lead to different normalizations.

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 content is not the same, however: rather than individual security, it is the security and continuing existence of an `` ideological group '' -- those in the `` free world '' -- that is basic.

Historically, however, the concept is one that has been of marked benefit to the people of the Western civilizational group.

The music director of the Pittsburgh Symphony Orchestra, William Steinberg, has molded his group into a prominent musical organization, which is his life.

With the group of boys it is different.

Nothing is more revealing of the way of life and literary aspirations of this group than their attitude toward sex.

This group is secularist and their program tends to be technological.

But the problem is one which gives us the measure of a man, rather than a group of men, whether a group of doctors, a group of party members assembled at a dinner to give their opinion, or the masses of the voters.

The group conducting the review is not holding formal hearings.

After allowing for group exposures, it is apparent that other factors must be considered if we are to comprehend fanaticism.

At Sounion there is a group of beautiful columns, the ruins of a temple to Poseidon, of particular interest at that time, as active reconstruction was in progress.

He is uncompromising in assigning guilt to the man who finds it necessary to inflict or permit injury to one individual or group for the sake of a larger good.

In the main stream of historical thinking is a group of scholars, H.M. Chadwick, R.H. Hodgkin, Sir Frank Stenton et al. who are in varying degrees sceptical of the native traditions of the conquest but who defend the catastrophic type of invasion suggested by them.

Although Patchen has given previous evidence of an interest in jazz, the musical group that he works with, the Chamber Jazz Sextet, is often ignored by jazz critics.

Patchen's musicians are outsiders in established jazz circles, and Patchen himself has remained outside the San Francisco poetry group, maintaining a self-imposed isolation, even though his conversion to poetry-and-jazz is not as extreme or as sudden as it may first appear.

Gov. Dalton's New Commerce and Industry Commission is moving to create a nine-state regional group in a collective effort to attract new industry.

Fortunately, there is a nursery school which he has been able to attend, with a group of normal children.

Perhaps Khrushchev is in a more difficult position than any since 1957, when the `` anti-party group '' nearly liquidated him.

When the fate of the individual is visited on the group, then ( the warm sweet butter dripped from her raised trembling fork and she pushed her head forward belligerently ), ah, then the true bitterness of existence could be tasted.

Mr. Claude is a specialist in torso development and he has long favored the now-famous Weider Push-Pull Super-Set technique in which one exercise of the Super-Set is a pressing or `` pushing '' movement which accents one sector of a muscle group in a specific way, followed by a `` pulling '' exercise which works the opposing sector of the same muscle group.

And all the time, she had the heat of hatred in her, like charcoal that is burning on its under side, but not visibly.

There was a measure of protection in its concrete walls and ceiling, but the engineers who hastily installed it were well aware that concrete is not much better than prayer, if as efficacious, when a direct hit comes along.

The long-settled areas of states like Virginia and South Carolina developed the ante-bellum culture to its richest flowering, and there the memory is more precious, and the consciousness of loss the greater.

As it is, they consider that the North is now reaping the fruits of excess egalitarianism, that in spite of its high standard of living the `` American way '' has been proved inferior to the English and Scandinavian ways, although they disapprove of the socialistic features of the latter.

Reduced to its simplest terms, it is an assumption of a collective duty to compensate for the inability of individuals to cope with the rigors of the era.

It is one of the ironic quirks of history that the viability and usefulness of nationalism and the territorial state are rapidly dissipating at precisely the time that the nation-state attained its highest number ( approximately 100 ).

While sovereignty has roots in antiquity, in its present usage it is essentially modern.

There is unceasing pressure, but its sources are immediate.

Others are confined to vast reservations, and not only does the Australian government justifiably not wish them to be viewed as exhibits in a zoo, but on their reservations they are extremely fugitive, shunning camps, coming together only for corroborees at which their strange culture comes to its highest pitch -- which is very low indeed.

But the South is, and has been for the past century, engaged in a wide-sweeping urbanization which, oddly enough, is not reflected in its literature.

All but the most rabid of Confederate flag wavers admit that the Old Southern tradition is defunct in actuality and sigh that its passing was accompanied by the disappearance of many genteel and aristocratic traditions of the reputedly languid ante-bellum way of life.

The answers derived by these means may determine not only the temporal organization of the dance but also its spatial design, special slips designating the location on the stage where the movement is to be performed.

The other variables include the dancer who is to perform the movement and the length of time he is to take in its performance.

Most of these, with horrible exceptions, were conceived as is a ship, not as an attempt to quell the ocean of mankind, nor to deny its force, but as a means to survive and enjoy it.

Hemingway's fiction is supported by a `` moral '' backbone and in its search for ultimate meaning hints at a religious dimension.

In addition, they have been converted to Zen Buddhism, with its glorification of all that is `` natural '' and mysteriously alive, the sense that everything in the world is flowing.

Jazz is good not only because it promotes wholeness but because of its decided sexual effect.

This confession serves to make clear in part what is behind this sexual revolution: the craving for sensation for its own sake, the need for change, for new experiences.

What one actually remembers is its greenness.

the mill-pond is quiet, its surface dark and shadowed, and there does not seem to be much water in it.

But the highroad, according to the description of its traffic, belongs to life as it is lived in unawareness of death, while the way to the churchyard belongs to some other sort of life: a suffering form, an existence wholly comprised in the awareness of death.