# 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.

# Lie Groups/Algebras, Notes 1: Lie Algebras

**Definition 1.** A **Lie algebra** is an -algebra such that and

Hence provided is not a ring of characteristic Also, if is an -algebra, then we can define a Lie algebra whose product is defined by

Hence in particular, simply doubles the product of Recall that a derivation on an algebra satisfies

and

For let us define the **adjoint** of as defined by

**Proposition 2.** For is a derivation on

*Proof.* By tensorial properties of algebras we have

And

**Proposition 3.** Let be a commutative Lie algebra over a ring with Then If then is commutative, regardless of the ring.

*Proof.* If is commutative, then The second claim is trivial.

Let be a left ideal in and Then and hence So is also a right ideal. Hence we simply consider ideals of Lie algebras without reference to leftness or rightness.

**Definition 4.** Let be a submodule of such that for all derivations on then is called a **characteristic ideal** of

It’s easy to see that a characteristic ideal is actually an ideal in since we have

for all

**Proposition 5.** Let be an ideal of and be a characteristic ideal of then is an ideal of

*Proof.* If is a derivation on then since is a characteristic ideal. But multiplication by is a derivation on Hence for

For two ideals we have that and are ideals in and that and In the case of an -algebra , we will let denote the submodule generated by elements This is in fact an ideal in a Lie algebra for if is a derivation, then

Hence it is stable under adjoint action.

**Definition 6.** The ideal is called the** derived ideal** of Inductively we can define

The sequence is called the **derived series** of Let us define and

The sequence is called the **lower central series** of

**Definition 7.** Let be a Lie algebra, be its tensor algebra (with interpreted as a module), and be the ideal generated by elements of the form Then is called the **enveloping algebra** of

**Theorem 8.** The enveloping algebra satisfies the universal property that if is an algebra homomorphism for some associative unital algebra and is the canonical inclusion, then there exists a unique morphism and such that

**Proposition 9.** If and are Lie algebras, then

Let us define with defined as above, and It follows that is a filtered algebra, which we will call the **associated filtered algebra** to

**Theorem 10 (Poincare-Birkhoff-Witt)** Let be the associated filtered algebra to and be the symmetric algebra of as a module. Then if is free,

It follows that if is free, then the canonical homomorphism is injective.

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

# Commutative Algebra, Notes 6: Valuations

**Definition 1.** Let be a field. An **absolute value** on is a map such that for all

- and iff

We will call the pair an **absolute field**. The absolute value that sends all nonzero elements to is called the trivial absolute value. One can also verify that the function defines a metric on Two absolute values are **dependent** if their induced topologies are the same, and **independent** otherwise. Note that

and

Hence and thus since Similarly

and

**Proposition 2.** Let and be two nontrivial absolute values on Then and are dependent iff If they are dependent, then there exist some such that

**Theorem 3. (Approximation Theorem) (Artin-Whaples)**. Let be a field and be nontrivial pairwise independent absolute values on Then there exist elements such that for any there exists an such that

for all

An absolute field is **complete** if all of its Cauchy sequences converge to a point in the field. One can show that each absolute field has a unique completion field up to isometry.

**Definition 4.** An abelian group is an **ordered group** if it has a partial ordering such that for all

**Proposition 5.** is an ordered group iff for a multiplicative subset where for all and

For an ordered group we will hereafter use to denote where and for all

**Definition 6.** Let be a field and be an ordered group. A **valuation** on is a map such that

- iff

We will call the triple a **valuation field**. The image of on is an ordered subgroup of Two valuations are **equivalent** if there is an isomorphism that respects order and value. One can also verify that and that if then

**Definition 7.** A subring of a field is a **valuation ring** if for all or

**Proposition 8.** If is a valuation ring, then it induces a valuation field.

*Proof.* Note that is a local ring, for, if are not units in then if we have

So if is a unit in , then which is a contradiction. Hence is not a unit. Furthermore if is not a unit, then for all we have that is not a unit. So the nonunits form an ideal in which is necessarily maximal (and uniquely so). So is a local ring. Let us denote this ideal by Thus we can write

where is the group of units of We now give a valuation. We will define our group as the quotient group In turn, we define with When then define iff and iff Observe that

and

Hence is a valuation field.

We thus hereafter refer to a *valuation ring* as the restriction of the induced valuation field to the ring.

**Proposition 9.** Let be fields where is a valuation field. Then there exists an extension making a valuation field.

*Proof Idea.* Let and We will call the **valuation ring** of One first takes the morphism and extends it to a valuation ring in One can construct an order preserving monomorphism

where is the maximal ideal in We then define as before, which is seen to agree with

**Proposition 10.** Let be fields and Let be a -valued valuation field and be the extension group to the induced valuation on Then

We will call the **ramification index** of in

**Definition 11.** A valuation is **discrete** if its codomain is a cyclic group.

It turns out that if is a valuation field and is the maximal ideal of its valuation ring, then there exists an element such that is a generator of Such an element is called a **local parameter** of One can also show that is a principal ideal generated by Moreover every element can be written as

for some unit integer is called the **order** of at If we say has a **zero** of order and if then we say has a **pole** of order

**Proposition 12.** Let be fields where is a finite extension of Further suppose that is a complete discrete valuation field (i.e. induced metric space is complete) and that and are the corresponding valuation rings and maximal ideals after extending the valuation. Then

[1] Lang, Serge. *Algebra*. Revised Third Edition. Springer-Verlag. 2000.