Below you will find example sentences with "lie group". The examples show how this phrase is used in natural context and which words often surround it.
Lie Group in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: lie
- Corpus frequency in the collocation scan: 29
- Phrase length: 2 words
- Average sentence length: 25.9 words
Sentence profile
- Phrase position: 7 start, 9 middle, 4 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "lie group" has 2 words and usually appears in the middle in these examples. The average sentence has 25.9 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as any connected lie group is a, groups a lie group is a, algebra, groups and connected stand out.
- In the phrase index, this combination connects with group equities, lie algebra, lie groups, lie algebra, lie groups and lie algebras, linking the page to nearby combinations.
Example types with lie group
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
Any Lie group gives rise to a Lie algebra. (9 words)
So every abstract Lie algebra is the Lie algebra of some (linear) Lie group. (14 words)
Lie groups A Lie group is a group in the category of smooth manifolds. (14 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)
The exponential map (Lie theory) from the Lie algebra to the Lie group, : provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. (40 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 exponential map (Lie theory) from the Lie algebra to the Lie group, : provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group.
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.
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.
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.
For every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.
Constructions There are several standard ways to form new Lie groups from old ones: *The product of two Lie groups is a Lie group.
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 ).
Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor.
So every abstract Lie algebra is the Lie algebra of some (linear) Lie group.
The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones.
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.
Lie groups A Lie group is a group in the category of smooth manifolds.
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.
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.
There is also a Lie group named E 7½ of dimension 190, but it is not a simple Lie 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.
Any Lie group gives rise to a Lie algebra.
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.
For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group.
Lie group method From 1870 Sophus Lie 's work put the theory of differential equations on a more satisfactory foundation.