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

Connected Lie in a sentence

Corpus data

  • Displayed example sentences: 6
  • Discovered as a combination around: lie
  • Corpus frequency in the collocation scan: 8
  • Phrase length: 2 words
  • Average sentence length: 25.2 words

Words in this phrase

connected lie

Meanings of words in this phrase

  • connected can mean: Having favorable rapport with a powerful entity. Frequent word
  • lie can mean: To rest in a horizontal position on a surface. Frequent word

Sentence profile

  • Phrase position: 3 start, 2 middle, 1 end
  • Sentence types: 6 statements, 0 questions, 0 exclamations

Usage patterns

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

a corresponding connected lie group unique 1× a simply connected lie group and 1× a simply connected lie group by 1× conversely any connected lie group is 1× corresponding simply connected lie groups 1× is a connected lie group of 1× of any connected lie group is 1× that any connected lie group with 1×

Sentence openings and endings

Common sentence openings:

as for classification 1× conversely to any 1× moreover every homomorphism 1× the universal cover 1× thus compact connected 1×

Common sentence endings:

been completely classified 1× connected lie groups 1× discrete central subgroup 1× of the center 1× s third theorem 1×

Context words

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

group 9× simply 3× universal 2× cover 2× conversely 2× discrete 2× subgroup 2× algebra 2× corresponding 2× unique 2× dimension 2× homomorphism 2×

Related phrases

lie algebra lie group lie groups lie algebras simply connected

Phrases around this combination

Other phrases discovered around lie:

lie algebra 20× lie group 20× lie groups 20× lie algebras 16×

Just before it in the phrase index:

connect people congressional republicans congressional hearing congressional district congressional democrats congressional delegation

Just after it in the phrase index:

connecting people consecutive games consecutive year consecutive years consensus price consensus rating

Corpus analysis

  • The phrase "connected lie" has 2 words and usually appears near the start in these examples. The average sentence has 25.2 words and is mostly made up of statements.
  • Around this phrase, patterns and context words such as a corresponding connected lie group unique, a simply connected lie group and, group, simply and universal 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 connected lie

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

Short example sentences

Thus, compact connected Lie groups have been completely classified. (9 words)

Moreover, every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups. (18 words)

Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering ( Lie's third theorem ). (27 words)

Longer example sentences

The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. (37 words)

It is a connected Lie group of dimension 2n 2 + n. *The group of invertible upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2. (32 words)

As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. (28 words)

Example sentences (6)

The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center.

Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering ( Lie's third theorem ).

As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup.

It is a connected Lie group of dimension 2n 2 + n. *The group of invertible upper triangular n by n matrices is a solvable Lie group of dimension n(n + 1)/2.

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

Thus, compact connected Lie groups have been completely classified.