# Fractional Sobolev Spaces

Recall for a function for we define the **Fourier transform** of by

and the **inverse Fourier transform** by

If then a.e..

Now if and with we say is the **weak derivative** of provided

with and

The **Sobolev space** is then defined as the set of all functions such that for all These become Banach spaces under the norm

and Hilbert when whence we denote

For we can say that consists of such that

(using Fourier multipliers).

**Bessel Potential Spaces**:

We can generalize with and and define the **Bessel potential space** as all such that Note the fractional power of the Laplacian is defined by virtue of the fact that is dense in , so we say

These spaces satisfy our desire of being Banach (and Hilbert when ).

**Sobolev-Slobodeckij Spaces:**

There is an alternative approach. Recall that the **Holder space** is defined as all functions such that

That is, it is the set of functions on which are and whose -th partial derivatives are bounded and Holder continuous of degree These spaces are Banach under the above norm. We can generalize the Sobolev spaces to incorporate similar properties. Let us define the **Slobodeckij norm** for with and by

The corresponding **Sobolev-Slobodeckij space** is defined as all functions such that

where This becomes a Banach space under the norm

[1] http://en.wikipedia.org/wiki/Sobolev_space (unclear text references)

[2] Lieb, Elliot and Michael Loss. *Analysis*. 2nd Edition. Graduate Studies in Mathematics. Vol. 14. American Mathematical Society. 2001.

[3] Evans, Lawrence. *Partial Differential Equations*. Graduate Studies in Mathematics. Vol. 19. American Mathematical Society. 1998.

# Lie Groups/Algebras, Notes 4: Lie Groups

**Lie Groups:**

**Definition 1.** A **Lie group** is a manifold and group whose binary operation is a map between the manifolds and

We will assume our Lie groups are Banach manifolds unless otherwise stated (locally homeomorphic to a Banach space ). It turns out a Lie group is metrizable, and the metric space is complete. Lie subgroups are subgroups which are also submanifolds and Lie groups.

**Proposition 2.** Let be an -Lie group (that is, a Lie group which is an -manifold). Then

- If then is locally connected.
- If then
- Suppose is locally compact, then is locally compact iff
- If is generated by a subspace whose topology admits a countable base, then the topology on admits a countable base.

Now suppose is a -manifold with an analytic associative binary operation with identity. Then the subset of invertible elements of is open in and is a Lie group.

**Definition 3.** A **Lie group morphism** between Lie groups and is a group homomorphism will denote the group of Lie group automorphisms of A **Lie group representation** on a Banach space is a Lie group morphism (Note if is a Banach space over the field , then is a Lie group which is a manifold over )

**Definition 4.** Let be a topological group and be an analytic manifold. We say that is a **homogeneous space** for if is a -space (that is, continuous action) and

This induces an equivalence relation on into orbits under the action of and a corresponding quotient space called the **orbit space** of by

**Proposition 5.** Let be Lie group and be a Lie subgroup. Then:

- The orbit space has a unique analytic manifold structure such that the canonical map is a submersion.
- If then is a Lie group and is a Lie group morphism.

In case 1 above, we call the quotient the **Lie homogeneous space** of by and in case 2 we call the quotient the **Lie quotient group**.

Let and be Lie groups. Then there is a canonical correspondence between and In particular, suppose we have two tangent vectors and Let us then define

It follows that for

Thus we have a product on the tangent bundle of (note that being a group makes an arbitrary element of ). Hence is a Lie group.

**The Lie Algebra of a Lie Group:**

Recall a vector field on a manifold is section of the vector fiber bundle with as its base. Hence it is a map satisfying where is the projection of the vector fiber bundle to its base. Now suppose and are two vector fields on then we can define the vector fields

where This turns the set of vector fields of into a Lie algebra.

Now let be a Lie group and be left multiplication by These are certainly diffeomorphisms of

**Definition 6.** A vector field is **left-invariant** if

for all where

**Proposition 7.** The set of left-invariant vector fields of is a Lie subalgebra of

*Proof.* Let and be left-invariant and . Then

and

The Lie subalgebra of is called the **Lie algebra of** Note an obvious consequence is that if and Then

Hence it follows that left-invariant vector fields of are completely determined by where they send the identity element. In this sense, we obtain the equivalence of the vector spaces and

**Lie Group of a Lie Algebra:**

The standard convention is to discuss an “inverse” map from the Lie algebra of a Lie group back to the Lie group. Let Then this left-invariant vector field corresponds to a tangent vector Let be a path defined by

Note these uniquely define called a **one-parameter subgroup** of We in turn define the map by

or more generally

Hence It follows that

Suppose Then and

[1] Bourbaki, Nicholas. *Lie Groups and Lie Algebras, Chapters 1-3*. Springer-Verlag. 1971.

[2] Robert Milson, Thomas Foregger, Mike Fikes. “Lie group” (version 12). *PlanetMath.org*. Freely available at http://planetmath.org/?op=getobj;from=objects;id=1112

# Lie Groups/Algebras, Notes 3: Bialgebras and Free Lie Algebras

Recall the construction of an associative algebra over a field We have a module homomorphism such that

If the algebra is unital, then we also a canonical map with defined as the identity of and corresponding assumption that

on (with canonical identifications ).

We can dually define a –**coalgebra** as a -module together with a **comultiplication map** such that

We also call it **unital** if there is a map such that

on (with the canonical identification ).

**Definition 1.** A –**bialgebra** is a unital -algebra and unital -coalgebra such that

- (),
- and

**Definition 2.** An element of a coalgebra is –**primitive** if Elements are **primitive** if they are 1-primitive.

If is the embedding into the enveloping algebra, then the map gives a -bialgebra structure.

**Definition 3.** Let be a set and be the free algebra over We define the **free Lie ****–****algebra over** as the quotient where is the ideal generated by elements of the form and We will denote this Lie algebra by and its product by

The enveloping algebra of the free Lie algebra ends up being the free associative algebra over

[1] Bourbaki, Nicholas. *Lie Groups and Lie Algebras, Chapters 1-3*. Springer-Verlag. 1971.

# Lie Groups/Algebras, Notes 2: Representations and Types

Recall a representation of an algebra on an -module is an algebra morphism When we have a Lie algebra a **representation** of it on an -module is an algebra morphism

Hence in particular we have We correspondingly call a –**module**. A representation is **faithful** if it is injective. It is **simple/irreducible** if its -module is simple.

Let and be representations of Lie algebras and on and respectively. Then is a representation of on defined by

where are the canonical inclusions of in and in Hence is both a -module and a -module.

**Definition 1**. is **solvable** if its derived series for some It is** nilpotent** if its lower central series for some The **radical** of is the largest solvable ideal in We say is **semisimple** if it has no nonzero abelian ideals.

Hence it follows that the center of a nilpotent Lie algebra is nontrivial. Also note that a nilpotent Lie algebra is solvable. is the smallest ideal such that has radical

**Theorem 2 (Engel)**. Let be a vector space and be a finite dimensional subalgebra of . If then there exists a with such that for all

It follows that a Lie algebra is nilpotent iff is nilpotent for all

Note that defines a symmetric bilinear form on called the **Killing form** of The Killing form also satisfies

**Proposition 3.** is solvable iff

**Proposition 4.** is semisimple iff iff is nondegenerate.

**Theorem 5 (Weyl)**. Every finite dimensional representation of a semisimple Lie algebra is completely reducible (i.e. the -module is semisimple).

A Lie algebra is **simple** if its only ideals are and and noncommutative.

**Proposition 6.** is semisimple iff it is a product of simple algebras.

A Lie algebra is r**eductive** if its adjoint representation is semisimple. That is, it is reductive if it can be written where is semisimple and is abelian (since here we want as a module to be semisimple–so that as an algebra, it has an an abelian term). This also turns out to be equivalent to being semisimple.

**Definition 7.** The **nilpotent radical** of is the intersection of the kernels of all finite dimensional simple representations of

For an element in the nilpotent radical, it follows that is nilpotent. Now for any nilpotent endomorphism of an algebra, we have that

has a finite number of nonzero terms, and is hence well-defined. If we can write with semisimple and nilpotent, then we will call a **Levi subalgebra** of

**Theorem 8 (Levi-Malcev).** Every Lie algebra has a Levi subalgebra (that is, it can be written as above). Moreover, every other Levi subalgebra is the image of under for some in the nilpotent radical of

It turns out that a subalgebra of is a Levi subalgebra iff it is a maximal semisimple subalgebra.

**Theorem 9 (Ado).** Let be the largest nilpotent ideal in Then admits a finite dimensional faithful representation such that every element of is nilpotent.

[1] Bourbaki, Nicholas. *Lie Groups and Lie Algebras, Chapters 1-3*. Springer-Verlag. 1971.