Below you will find example sentences with "lie groups". The examples show how this phrase is used in natural context and which words often surround it.

Lie Groups in a sentence

Corpus data

  • Displayed example sentences: 20
  • Discovered as a combination around: lie
  • Corpus frequency in the collocation scan: 24
  • Phrase length: 2 words
  • Average sentence length: 24.4 words

Words in this phrase

lie groups

Meanings of words in this phrase

  • lie can mean: To rest in a horizontal position on a surface. Frequent word
  • groups can mean: plural of group Common word

Sentence profile

  • Phrase position: 11 start, 6 middle, 3 end
  • Sentence types: 20 statements, 0 questions, 0 exclamations

Usage patterns

These patterns show short text windows around the phrase in the example sentences:

algebras and lie groups is used 1× categories different lie groups may have 1× classification of lie groups and the 1× construction on lie groups on an 1× correspondence between lie groups and lie 1× defined for lie groups that are 1× definition for lie groups a lie 1× example simple lie groups are usually 1×

Sentence openings and endings

Common sentence openings:

a construction on 1× constructions there are 1× diffeomorphism groups of 1× for example simple 1× further advancing these 1× however even though 1×

Common sentence endings:

a lie group 2× of smooth manifolds 2× a matrix group 1× and lie algebras 1× any lie group 1× areas of study 1×

Context words

These words often appear near this phrase in the example sentences:

algebras 12× group 12× isomorphic 5× study 4× homomorphism 4× theory 3× dimensional 3× two 3× corresponding 3× category 3× manifolds 3× correspondence 2×

Related phrases

ethnic groups lie algebra lie group lie algebras religious groups terror groups minority groups terrorist groups community groups connected lie

Phrases around this combination

Other phrases discovered around lie:

lie algebra 20× lie group 20× lie algebras 16× connected lie 8×

Just before it in the phrase index:

lie group lie algebras lie algebra license plates license plate licence fee

Just after it in the phrase index:

lieutenant governor life cycle life insurance life lessons life magazine life saving

Other phrases with the same pattern

Same start:

lie algebra lie group lie algebras

Same ending:

ethnic groups religious groups terror groups minority groups terrorist groups community groups

Corpus analysis

  • The phrase "lie groups" has 2 words and usually appears near the start in these examples. The average sentence has 24.4 words and is mostly made up of statements.
  • Around this phrase, patterns and context words such as algebras and lie groups is used, categories different lie groups may have, algebras, group and isomorphic stand out.
  • In the phrase index, this combination connects with ethnic groups, lie algebra, lie group, lie algebra, lie group and lie algebras, linking the page to nearby combinations.

Example types with lie groups

This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:

Short example sentences

Lie's fundamental theorems describe a relation between Lie groups and Lie algebras. (13 words)

Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884. (14 words)

Lie groups A Lie group is a group in the category of smooth manifolds. (14 words)

Longer example sentences

This Lie algebra is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. (44 words)

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. (43 words)

It is however not an equivalence of categories : different Lie groups may have isomorphic Lie algebras (for example SO(3) and SU(2) ), and there are (infinite dimensional) Lie algebras that are not associated to any Lie group. (38 words)

Example sentences (20)

The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups.

This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras.

It is however not an equivalence of categories : different Lie groups may have isomorphic Lie algebras (for example SO(3) and SU(2) ), and there are (infinite dimensional) Lie algebras that are not associated to any Lie group.

Quotients of Lie groups If G is a Lie group and N is a normal Lie subgroup of G, the quotient G / N is also a Lie group.

Constructions There are several standard ways to form new Lie groups from old ones: *The product of two Lie groups is a Lie group.

Relation to Lie groups seeAlso Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups.

Lie's fundamental theorems describe a relation between Lie groups and Lie algebras.

Lie theory with its Lie groups and Lie algebras became one of the major areas of study.

Advertentie

This Lie algebra is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element.

Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group.

A construction on Lie groups On an n-dimensional Lie group, Haar measure can be constructed easily as the measure induced by a left-invariant n-form.

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

Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.

However, even though su(2) and so(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups.

Lie groups A Lie group is a group in the category of smooth manifolds.

Moreover, every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups.

The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds.

Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism.

Diffeomorphism groups of compact manifolds of larger dimension are regular Fréchet Lie groups ; very little about their structure is known.

Advertentie