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

Lie Algebras in a sentence

Corpus data

  • Displayed example sentences: 20
  • Discovered as a combination around: lie
  • Corpus frequency in the collocation scan: 16
  • Phrase length: 2 words
  • Average sentence length: 19.7 words

Words in this phrase

lie algebras

Meanings of words in this phrase

  • lie can mean: To rest in a horizontal position on a surface. Frequent word
  • algebras can mean: plural of algebra Less frequent word

Sentence profile

  • Phrase position: 5 start, 9 middle, 6 end
  • Sentence types: 19 statements, 1 questions, 0 exclamations

Usage patterns

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

the corresponding lie algebras 2× applied to lie algebras 1× correspondence between lie algebras and lie 1× determined simple lie algebras 1× five exceptional lie algebras that do 1× for lie algebras the definition 1× for the lie algebras and the 1× groups and lie algebras 1×

Sentence openings and endings

Common sentence openings:

conversely it can 1× for example simple 1× for lie algebras 1× furthermore some infinite 1× hence the question 1× however even though 1×

Common sentence endings:

corresponding lie algebras 2× accomplished in general 1× and associative algebras 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:

groups 16× group 4× theory 4× corresponding 4× homomorphism 4× isomorphic 3× study 3× associative 3× algebra 3× simple 3× different 2× example 2×

Related phrases

lie algebra lie group lie groups connected lie

Phrases around this combination

Other phrases discovered around lie:

lie algebra 20× lie group 20× lie groups 20× connected lie 8×

Just before it in the phrase index:

lie algebra license plates license plate licence fee library system library staff

Just after it in the phrase index:

lie group lie groups lieutenant governor life cycle life insurance life lessons

Other phrases with the same pattern

Same start:

lie algebra lie group lie groups

Corpus analysis

  • The phrase "lie algebras" has 2 words and usually appears in the middle in these examples. The average sentence has 19.7 words and is mostly made up of statements.
  • Around this phrase, patterns and context words such as the corresponding lie algebras, applied to lie algebras, groups, group and theory stand out.
  • In the phrase index, this combination connects with lie algebra, lie group, lie groups, lie algebra, lie group and lie groups, linking the page to nearby combinations.

Example types with lie algebras

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

Short example sentences

For Lie algebras the definition is slightly different. (8 words)

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

Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. (13 words)

Longer example sentences

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)

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. (31 words)

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. (26 words)

Question examples

Hence the question arises: what are the simple Lie algebras of compact groups? (13 words)

Example sentences (20)

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.

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

There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.

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.

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

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.

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.

Advertentie

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

Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras.

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

Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

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

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

Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras.

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.

But there are also just five "exceptional Lie algebras" that do not fall into any of these families.

For Lie algebras the definition is slightly different.

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

However, the classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.

Advertentie