aarjaneiro/planetmath huggingface dataset:
groupoid action

groupoid action

Definition 0.1

.

Let

𝒢

be a

groupoid

and

X

a

topological space

. A

groupoid action

, or

𝒢

-action, on

X

is given by two maps: the

anchor map

π

:

X

⟶

G

0

and a map

μ

:

X

×

G

0

G

1

⟶

X

,

with the latter being defined on pairs

(

x

,

g

)

such that

π

⁢

(

x

)

=

t

⁢

(

g

)

, written as

μ

⁢

(

x

,

g

)

=

x

⁢

g

. The two maps are subject to the following conditions:

•

π

⁢

(

x

⁢

g

)

=

s

⁢

(

g

)

,

•

x

⁢

u

⁢

(

π

⁢

(

x

)

)

=

x

,

and

•

(

x

⁢

g

)

⁢

h

=

x

⁢

(

g

⁢

h

)

,

whenever the operations are defined.

Note:

The groupoid action generalizes the concept of group action in a non-trivial way.

Title

groupoid action

Canonical name

GroupoidAction

Date of creation

2013-03-22 19:19:23

Last modified on

2013-03-22 19:19:23

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

9

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22A22

Classification

msc 18B40

Synonym

action

Related topic

GroupAction

Related topic

Groupoid

Related topic

GroupoidRepresentation4

Defines

anchor map

Generated on Fri Feb 9 19:39:32 2018 by

LaTeXML
Polish G-space

Polish G-space

Definition 0.1

.

Let

X

be a topological

G

-space (

http://planetmath.org/TopologicalGSpace

), and

G

its associated

topological group

,
that is, such that an action

a

of

G

on

X

is

continuous

if

a

:

G

×

X

→

X

is continuous.
If

G

is a

Polish group

and

X

is also a

Polish space

, then

X

is called a

Polish G-space

.

References

1

Howard Becker, Alexander S. Kechris. 1996.

The

Descriptive Set Theory

of Polish Group Actions

Cambridge University Press: Cambridge, UK, p.14.

Title

Polish G-space

Canonical name

PolishGspace

Date of creation

2013-03-22 18:24:37

Last modified on

2013-03-22 18:24:37

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

9

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22A22

Classification

msc 22A10

Classification

msc 22A05

Classification

msc 54H05

Related topic

TopologicalGSpace

Related topic

PolishGroup

Related topic

Group

Related topic

TopologicalGroup2

Related topic

PointedTopologicalSpace

Related topic

BasicResultsInTopologicalGroups

Generated on Fri Feb 9 19:39:34 2018 by

LaTeXML
small category- alternative

small category- alternative

Definition 0.1

.

A (small)

category

𝒞

consists of a set of objects

C

0

and a set of arrows

C

1

together with the following

structure

:

•

a source map

s

:

C

1

→

C

0

assigning an object

s

⁢

(

f

)

to each arrow

f

∈

C

1

,

•

a target map:

t

:

C

1

→

C

0

assigning an object t(f) to each arrow

f

∈

C

1

,

•

an

identity map

1

:

C

0

→

C

1

assigning to each object

A

an arrow

1

A

with

s

⁢

(

1

A

)

=

t

⁢

(

1

A

)

=

A

,

•

a

composition

map

∘

:

C

1

×

C

1

→

C

1

assigning to each pair of arrows

f

,

g

, such that

s

⁢

(

g

)

=

t

⁢

(

f

)

, a third arrow

g

∘

f

with

s

⁢

(

g

∘

f

)

=

s

⁢

(

f

)

and

t

⁢

(

g

∘

f

)

=

t

⁢

(

g

)

.

•

The composition thus defined “

∘

” is associative, that is,

h

∘

(

g

∘

f

)

=

(

h

∘

g

)

∘

f

whenever these compositions make sense.

•

the identity map

satisfies

f

∘

1

A

=

f

for any

f

such that

s

⁢

(

f

)

=

A

and

1

A

∘

g

=

g

, and any

g

such that

t

⁢

(

g

)

=

A

.

References

P. A. Zito. 2008. [arXiv: math.CT].

http://arxiv.org/PS_cache/math/pdf/0509/0509266v1.pdf

2-

C

*

-Categories with non-simple units.
,(Preprint).

Title

small category- alternative

Canonical

name

SmallCategoryAlternative

Date of creation

2013-03-22 18:26:38

Last modified on

2013-03-22 18:26:38

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22A22

Classification

msc 18E05

Classification

msc 18-00

Related topic

Category

Related topic

2CCategory

Generated on Fri Feb 9 19:39:36 2018 by

LaTeXML
Polish group

Polish group

Definition 0.1

.

Let

G

be a

topological group

regarded as a

topological space

.
Then

G

is defined to be a

Polish group

if it is also a

Polish space

.

References

1

Howard Becker, Alexander S. Kechris. 1996.

The

Descriptive Set Theory

of Polish Group Actions

Cambridge University Press: Cambridge, UK, p.14.

Title

Polish group

Canonical name

PolishGroup

Date of creation

2013-03-22 18:24:40

Last modified on

2013-03-22 18:24:40

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

5

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22A25

Classification

msc 22A22

Classification

msc 22A10

Classification

msc 54H05

Classification

msc 22A05

Related topic

PolishSpace

Related topic

Group

Related topic

PolishGSpace

Related topic

TopologicalGspace

Related topic

TopologicalGroup2

Related topic

BasicResultsInTopologicalGroups

Related topic

TopologicalGSpace

Generated on Fri Feb 9 19:39:37 2018 by

LaTeXML
topological *-algebra

topological

*

-algebra

Definition

(Involution)

An involution on an

algebra

A

over an

involutory

field (

http://planetmath.org/InvolutaryRing

)

F

is a map

⋅

*

:

A

→

A

:

a

↦

a

*

such that for every

{

a

,

b

}

⊂

A

and

λ

∈

F

we have

1.

a

*

*

=

a

,

2.

(

a

⁢

b

)

*

=

b

*

⁢

a

*

and

3.

(

λ

⁢

a

+

b

)

*

=

λ

*

⁢

a

*

+

b

*

, where

λ

*

denotes the involution (

http://planetmath.org/InvolutaryRing

) of

λ

in

F

.

Definition (

*

-Algebra)

A

*

-algebra is an algebra with an involution.

Definition (Topological

*

-algebra)

A topological

*

-algebra is a

*

-algebra which is also a

topological vector space

such that its algebra

multiplication

and involution are

continuous

.

0.0.1

Remarks:

•

*

-algebras are a particular of involutory rings.

•

The involutory field

F

is often taken as

ℂ

, where the involution is given by

complex conjugation

. In this case, condition 3 could be rewritten as:

3.

(

λ

⁢

a

+

b

)

*

=

λ

¯

⁢

a

*

+

b

*

•

Banach *-algebras are topological

*

-algebras.

Title

topological

*

-algebra

Canonical

name

Topologicalalgebra

Date of creation

2013-03-22 14:45:38

Last modified on

2013-03-22 14:45:38

Owner

HkBst (6197)

Last modified by

HkBst (6197)

Numerical id

12

Author

HkBst (6197)

Entry type

Definition

Classification

msc 22A30

Classification

msc 16W80

Classification

msc 16W10

Classification

msc 46K05

Classification

msc 46H35

Synonym

topological *-algebra

Related topic

BanachAlgebra

Related topic

WeakHopfCAlgebra2

Related topic

VonNeumannAlgebra

Defines

involution

*

-algebra

Defines

*-algebra

Generated on Fri Feb 9 19:39:39 2018 by

LaTeXML
compact groups are unimodular

compact groups are unimodular

Theorem

-

If

G

is a

compact

Hausdorff

topological group

, then

G

is unimodular, i.e. it’s left and right

Haar measures

coincide.

Proof

:

Let

Δ

denote the

modular function

of

G

. It is enough to prove that

Δ

is

constant

and equal to

1

, since this proves that every left Haar measure is right

invariant

.

Since

Δ

is

continuous

and

G

is compact,

Δ

⁢

(

G

)

is a compact subset of

ℝ

+

. In particular,

Δ

⁢

(

G

)

is a

bounded

subset of

ℝ

+

.

But if

Δ

is not identically one, then there is a

t

∈

G

such that

Δ

⁢

(

t

)

>

1

(recall that

Δ

is an

homomorphism

). Hence,

Δ

⁢

(

t

n

)

=

Δ

⁢

(

t

)

n

⟶

∞

as

n

∈

ℕ

increases, which is a

contradiction

since

Δ

⁢

(

G

)

is bounded.

□

Title

compact groups are unimodular

Canonical name

CompactGroupsAreUnimodular

Date of creation

2013-03-22 17:58:23

Last modified on

2013-03-22 17:58:23

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 22C05

Classification

msc 28C10

Generated on Fri Feb 9 19:39:41 2018 by

LaTeXML
n-torus

n

-torus

The

n

-torus

, denoted

T

n

, is a smooth orientable

n

dimensional manifold which is the product of

n

1-spheres, i.e.

T

n

=

S

1

×

⋯

×

S

1

⏟

n

.

Equivalently, the

n

-torus can be considered to be

ℝ

n

modulo the action (

vector addition

) of the

integer lattice

ℤ

n

.

The

n

-torus is in

addition

a

topological group

. If we think of

S

1

as the unit circle in

ℂ

and

T

n

=

S

1

×

⋯

×

S

1

⏟

n

, then

S

1

is a topological group and so is

T

n

by coordinate-wise

multiplication

. That is,

(

z

1

,

z

2

,

…

,

z

n

)

⋅

(

w

1

,

w

2

,

…

,

w

n

)

=

(

z

1

⁢

w

1

,

z

2

⁢

w

2

,

…

,

z

n

⁢

w

n

)

Title

n

-torus

Canonical name

Ntorus

Date of creation

2013-03-22 13:59:58

Last modified on

2013-03-22 13:59:58

Owner

ack (3732)

Last modified by

ack (3732)

Numerical id

7

Author

ack (3732)

Entry type

Definition

Classification

msc 22C05

Classification

msc 54B10

Related topic

Torus

Generated on Fri Feb 9 19:39:44 2018 by

LaTeXML
reductive

reductive

Let

G

be a Lie group or

algebraic group

.

G

is called

reductive

over a field

k

if every

representation

of

G

over

k

is

completely reducible

For example, a

finite group

is reductive over a field

k

if and only if its order is not divisible by the

characteristic

of

k

(by Maschke’s

theorem

). A complex Lie group is reductive if and only if it is a

direct product

of a

semisimple group

and an

algebraic torus

.

Title

reductive

Canonical name

Reductive

Date of creation

2013-03-22 13:23:49

Last modified on

2013-03-22 13:23:49

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Definition

Classification

msc 22C05

Generated on Fri Feb 9 19:39:45 2018 by

LaTeXML
Γ-simple

Γ

-simple

A

representation

V

of

Γ

is

Γ

-simple

if either

•

V

≅

W

1

⊕

W

2

where

W

1

,

W

2

are absolutely irreducible for

Γ

and are

Γ

-

isomorphic

, or

•

V

is non-absolutely

irreducible

for

Γ

.

[

GSS

]

References

GSS

Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory

(Volume II)

. Springer-Verlag, New York, 1988.

Title

Γ

-simple

Canonical name

Gammasimple

Date of creation

2013-03-22 13:52:11

Last modified on

2013-03-22 13:52:11

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 22D05

Generated on Fri Feb 9 19:39:47 2018 by

LaTeXML
modular function

modular function

Let

G

be a locally compact Hausdorff

topological group

and

μ

a

left Haar measure

. Although left and right Haar measures in

G

always exist, they generally do not coincide, i.e. a left Haar measure is usually not

invariant

under right translations. Nevertheless, the right translations of a left Haar measure can be easily described as explained in the following

theorem

.

Theorem -

Let

G

be a locally compact Hausdorff topological group and

μ

a left Haar measure in

G

. Then, there exists a

continuous

homomorphism

Δ

:

G

⟶

ℝ

+

such that, for every

t

∈

G

and every measurable subset

A

μ

⁢

(

A

⁢

t

)

=

Δ

⁢

(

t

-

1

)

⁢

μ

⁢

(

A

)

Moreover, if

f

:

G

⟶

ℂ

is an integrable function then

Δ

⁢

(

t

)

⁢

∫

G

f

⁢

(

s

⁢

t

)

⁢

μ

⁢

(

s

)

=

∫

G

f

⁢

(

s

)

⁢

μ

⁢

(

s

)

The

function

Δ

is called the

modular function

of

G

(notice that, by uniqueness up to scalar multiple of left Haar measures,

Δ

only depends on

G

). Other names for

Δ

that can be found are:

Haar modulus

, or

modular character

or

modular homomorphism

.

We now prove the above theorem, except the continuity of

Δ

(which is slightly harder to obtain).

Proof (except continuity of

Δ

):

Let

t

∈

G

. The function

ν

, defined on measurable subsets

A

by

ν

⁢

(

A

)

:=

μ

⁢

(

A

⁢

t

)

is easily seen to be a

measure

in

G

. Moreover,

ν

is left invariant (since

μ

is left invariant) and

satisfies

the additional conditions to be a left Haar measure. By the uniqueness of left Haar measures,

μ

must be a multiple of

ν

, i.e.

μ

=

Δ

⁢

(

t

)

⁢

ν

for some

positive

scalar

Δ

⁢

(

t

)

∈

ℝ

+

. Thus, we have proven that for every measurable subset

A

μ

⁢

(

A

⁢

t

)

=

Δ

⁢

(

t

)

-

1

⁢

μ

⁢

(

A

)

Now for

s

,

t

∈

G

we have that

μ

⁢

(

A

⁢

s

⁢

t

)

=

Δ

⁢

(

s

⁢

t

)

-

1

⁢

μ

⁢

(

A

)

, but also

•

μ

⁢

(

A

⁢

s

⁢

t

)

=

Δ

⁢

(

t

)

-

1

⁢

μ

⁢

(

A

⁢

s

)

, and

•

μ

⁢

(

A

⁢

s

)

=

Δ

⁢

(

s

)

-

1

⁢

μ

⁢

(

A

)

So, we can see that, for every measurable subset

A

,

Δ

⁢

(

s

⁢

t

)

-

1

⁢

μ

⁢

(

A

)

=

Δ

⁢

(

t

)

-

1

⁢

Δ

⁢

(

s

)

-

1

⁢

μ

⁢

(

A

)

Hence,

Δ

⁢

(

s

⁢

t

)

=

Δ

⁢

(

s

)

⁢

Δ

⁢

(

t

)

. Thus,

Δ

is an homomorphism.

The statement about

integrals

of functions follows easily by approximation by

simple functions

. For simple functions it is easy to see it is true using the now established condition

μ

⁢

(

A

⁢

t

)

=

Δ

⁢

(

t

-

1

)

⁢

μ

⁢

(

A

)

.

□

Title

modular function

Canonical

name

ModularFunction

Date of creation

2013-03-22 17:58:18

Last modified on

2013-03-22 17:58:18

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Definition

Classification

msc 22D05

Classification

msc 28C10

Synonym

Haar modulus

Synonym

modular character

Synonym

modular homomorphism

Generated on Fri Feb 9 19:39:50 2018 by

LaTeXML
group C^*-algebra

group

C

*

-algebra

Let

ℂ

⁢

[

G

]

be the

group ring

of a

discrete group

G

.
It has two

completions

to a

C

*

-

algebra

:

Reduced

group

C

*

-algebra.

The reduced group

C

*

-algebra,

C

r

*

⁢

(

G

)

, is obtained by completing

ℂ

⁢

[

G

]

in the

operator norm

for its

regular representation

on

l

2

⁢

(

G

)

.

Maximal

group

C

*

-algebra.

The maximal group

C

*

-algebra,

C

max

*

⁢

(

G

)

or just

C

*

⁢

(

G

)

,
is defined by the following

universal property

:
any *-

homomorphism

from

ℂ

⁢

[

G

]

to some

𝔹

⁢

(

ℋ

)

(the

C

*

-algebra of

bounded operators

on some

Hilbert space

ℋ

)
factors through the

inclusion

ℂ

⁢

[

G

]

↪

C

max

*

⁢

(

G

)

.

If

G

is

amenable

then

C

r

*

⁢

(

G

)

≅

C

max

*

⁢

(

G

)

.

Title

group

C

*

-algebra

Canonical

name

GroupCalgebra

Date of creation

2013-03-22 13:10:53

Last modified on

2013-03-22 13:10:53

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

6

Author

mhale (572)

Entry type

Definition

Classification

msc 22D15

Related topic

CAlgebra

Related topic

GroupoidCConvolutionAlgebra

Generated on Fri Feb 9 19:39:52 2018 by

LaTeXML
unimodular group

unimodular group

A

locally compact

Hausdorff

topological group

is said to be

if
its

left Haar measure

is equal to its right Haar measure.

A group is when its

modular function

is equal to 1.

For example, an

Abelian group

or a compact group or

discrete group

is .

Title

unimodular group

Canonical name

UnimodularGroup

Date of creation

2013-03-22 16:51:43

Last modified on

2013-03-22 16:51:43

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 22D99

Generated on Fri Feb 9 19:39:54 2018 by

LaTeXML
existence and uniqueness of compact real form

existence and uniqueness of compact real form

Let

G

be a

semisimple

complex Lie group. Then there exists a unique (up to

isomorphism

) real

Lie group

K

such that

K

is

compact

and a real form of

G

. Conversely, if

K

is compact, semisimple and real, it is the real form of a unique semisimple complex Lie group

G

. The group

K

can be realized as the set of

fixed points

of a special

involution

of

G

, called the

Cartan involution

.

For example, the compact real form of

SL

n

⁢

ℂ

, the complex

special linear group

, is

SU

⁢

(

n

)

, the

special unitary group

. Note that

SL

n

⁢

ℝ

is also a real form of

SL

n

⁢

ℂ

, but is

not

compact.

The compact real form of

SO

n

⁢

ℂ

, the complex

special orthogonal group

, is

SO

n

⁢

ℝ

, the real

orthogonal group

.

SO

n

⁢

ℂ

also has other, non-compact real forms, called the pseudo-orthogonal groups.

The compact real form of

Sp

2

⁢

n

⁢

ℂ

, the complex

symplectic group

, is less well-known. It is (unfortunately) also usually denoted

Sp

⁢

(

2

⁢

n

)

, and consists of

n

×

n

“

unitary

”

quaternion

matrices, that is,

Sp

⁢

(

2

⁢

n

)

=

{

M

∈

GL

n

⁢

ℍ

|

M

⁢

M

*

=

I

}

where

M

*

denotes

M

conjugate transpose

. This different from the real symplectic group

Sp

2

⁢

n

⁢

ℝ

.

Title

existence and uniqueness of compact real form

Canonical name

ExistenceAndUniquenessOfCompactRealForm

Date of creation

2013-03-22 13:23:37

Last modified on

2013-03-22 13:23:37

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Theorem

Classification

msc 22E10

Generated on Fri Feb 9 19:39:56 2018 by

LaTeXML
maximal torus

maximal torus

Let

K

be a compact group, and let

t

∈

K

be an element whose

centralizer

has minimal

dimension

(such elements are dense in

K

). Let

T

be the centralizer of

t

. This

subgroup

is closed since

T

=

φ

-

1

⁢

(

t

)

where

φ

:

K

→

K

is the map

k

↦

k

⁢

t

⁢

k

-

1

, and

abelian

since it is the intersection of

K

with the

Cartan subgroup

of its

complexification

, and hence a torus, since

K

(and thus

T

) is

compact

. We call

T

a

maximal torus

of

K

.

This term is also applied to the corresponding

maximal

abelian subgroup of a complex

semisimple group

, which is an algebraic torus.

Title

maximal torus

Canonical

name

MaximalTorus

Date of creation

2013-03-22 13:23:52

Last modified on

2013-03-22 13:23:52

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 22E10

Classification

msc 22C05

Generated on Fri Feb 9 19:40:01 2018 by

LaTeXML
O(2)

O(2)

still being written

An

elementary

example of a Lie group is afforded by O(2),
the

orthogonal group

in two

dimensions

. This is the set
of

transformations

of the plane which fix the origin and

preserve

the distance between points. It may be shown
that a transform has this

property

if and only if it is of
the form

(

x

y

)

↦

M

⁢

(

x

y

)

,

where

M

is a

2

×

2

matrix such that

M

T

⁢

M

=

I

.
(Such a matrix is called

orthogonal

.)

It is easy enough to check that this is a group. To see
that it is a Lie group, we first need to make sure that it
is a manifold. To that end, we will parameterize it.
Calling the entries of the matrix

a

,

b

,

c

,

d

, the condition
becomes

(

0

1

1

0

)

=

(

a

b

c

d

)

T

⁢

(

a

b

c

d

)

=

(

a

2

+

c

2

a

⁢

b

+

c

⁢

d

a

⁢

b

+

c

⁢

d

b

2

+

d

2

)

which is

equivalent

to the following system of equations:

a

2

+

c

2

=

1

a

⁢

b

+

c

⁢

d

=

0

b

2

+

d

2

=

1

The first of these

equations

can be solved by introducing a

parameter

θ

and writing

a

=

cos

⁡

θ

and

c

=

sin

⁡

θ

. Then the second equation becomes

b

⁢

cos

⁡

θ

+

d

⁢

sin

⁡

θ

=

0

, which can be solved by
introducing a parameter

r

:

b

=

-

r

⁢

sin

⁡

θ

d

=

r

⁢

cos

⁡

θ

Substituting this into the third equation results in

r

2

=

1

,
so

r

=

-

1

or

r

=

+

1

. This means we have two matrices for
each value of

θ

:

(

cos

⁡

θ

-

sin

⁡

θ

sin

⁡

θ

cos

⁡

θ

)

(

cos

⁡

θ

sin

⁡

θ

sin

⁡

θ

-

cos

⁡

θ

)

Since more than one value of

θ

will produce the same
matrix, we must restrict the range in order to obtain a
bona fide

coordinate

. Thus, we may cover

O

⁢

(

2

)

with an
atlas consisting of four

neighborhoods

:

{

(

cos

⁡

θ

-

sin

⁡

θ

sin

⁡

θ

cos

⁡

θ

)

∣

-

3

4

⁢

π

<

θ

<

3

4

⁢

π

}

{

(

cos

⁡

θ

-

sin

⁡

θ

sin

⁡

θ

cos

⁡

θ

)

∣

1

4

⁢

π

<

θ

<

7

4

⁢

π

}

{

(

cos

⁡

θ

sin

⁡

θ

sin

⁡

θ

-

cos

⁡

θ

)

∣

-

3

4

⁢

π

<

θ

<

3

4

⁢

π

}

{

(

cos

⁡

θ

sin

⁡

θ

sin

⁡

θ

-

cos

⁡

θ

)

∣

1

4

⁢

π

<

θ

<

7

4

⁢

π

}

Every element of

O

⁢

(

2

)

must belong to at least one of
these neighborhoods. It its

trivial

to check that the

transition functions

between overlapping coordinate
patches are

Title

O(2)

Canonical

name

O2

Date of creation

2013-03-22 17:57:38

Last modified on

2013-03-22 17:57:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Example

Classification

msc 22E10

Classification

msc 22E15

Generated on Fri Feb 9 19:40:04 2018 by

LaTeXML
complexification

complexification

Let

G

be a real Lie group. Then the

complexification

G

ℂ

of

G

is the unique complex Lie group equipped with a map

φ

:

G

→

G

ℂ

such that any map

G

→

H

where

H

is a
complex Lie group, extends to a holomorphic map

G

ℂ

→

H

. If

𝔤

and

𝔤

ℂ

are the respective Lie algebras,

𝔤

ℂ

≅

𝔤

⊗

ℝ

ℂ

.

For

simply connected

groups, the construction is obvious: we simply take the
simply connected complex

group with

Lie algebra

𝔤

ℂ

, and

φ

to be the
map induced by the

inclusion

𝔤

→

𝔤

ℂ

.

If

γ

∈

G

is central, then its image is in central in

G

ℂ

since

g

↦

γ

⁢

g

⁢

γ

-

1

is a map extending

φ

, and thus must be the

identity

by uniqueness half of the

universal property

. Thus, if

Γ

⊂

G

is a discrete central subgroup, then we get a map

G

/

Γ

→

G

ℂ

/

φ

⁢

(

Γ

)

, which gives a complexification for

G

/

Γ

. Since every Lie group is of this form, this shows

existence

.

Some easy examples: the complexification both of

SL

n

⁢

ℝ

and

SU

⁢

(

n

)

is

SL

n

⁢

ℂ

. The complexification of

ℝ

is

ℂ

and of

S

1

is

ℂ

*

.

The map

φ

:

G

→

G

ℂ

is not always

injective

. For example, if

G

is
the

universal cover

of

SL

n

⁢

ℝ

(which has

fundamental group

ℤ

), then

G

ℂ

≅

SL

n

⁢

ℂ

, and

φ

factors through the

covering

G

→

SL

n

⁢

ℝ

.

Title

complexification

Canonical name

Complexification

Date of creation

2013-03-22 13:53:55

Last modified on

2013-03-22 13:53:55

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 22E15

Generated on Fri Feb 9 19:40:06 2018 by

LaTeXML