1. Introduction
A Lie group is, roughly speaking, an analytic manifold with a group structure such that the group operations are analytic. Lie groups arise in a natural way as transformation groups of geometric objects. Lie groups are not linear they are curved manifolds. Nevertheless, Lie’s theorem reduces many questions about Lie groups to questions about Lie algebras. Questions about curved manifolds turn out to be equivalent to questions about linear algebra. This is a profound simplification, and it leads to a very rich theory. The theory of Lie groups answers these questions by replacing the notion of a finitely generated group with that of a Lie group—a group which at the same time is a finite-dimensional manifold. It turns out that in many ways such groups can be described and studied as easily as finitely generated groups—or even easier.
Lie groups and Lie algebras, together called Lie theory, originated in the study of natural symmetries of solutions of differential equations. However, unlike say the finite collection of symmetries of the hexagon, these symmetries occurred in continuous families, just as the rotational symmetries of the plane form a continuous family isomorphic to the unit circle. The theory as we know it today began with the ground-breaking work of the Norwegian mathematician Sophus Lie, who introduced the notion of continuous transformation groups and showed the crucial role that Lie algebras play in their classification and representation theory. Lie’s ideas played a central role in Felix Klein’s grand “Erlangen program” to classify all possible geometries using group theory. Today Lie theory plays an important role in almost every branch of pure and applied mathematics, is used to describe much of modern physics, in particular classical and quantum mechanics, and is an active area of research [1].
The General Linear Group GL(n, R) is the group of invertible n × n matrices with real entries under matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in a general linear position. https://en.wikipedia.org/wiki/General_linear_group
The special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant
where we write F× for the multiplicative group of F (that is, F excluding 0).
These elements are “special” in that they form a subvariety of the general linear group—they satisfy a polynomial equation (since the determinant is polynomial in the entries). https://en.wikipedia.org/wiki/Special_linear_group
The orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. An orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. The orthogonal group is an algebraic group and a Lie group. It is compact.
The orthogonal group in dimension n has two connected components. The one that contains the identity element is a subgroup, called the special orthogonal group, https://en.wikipedia.org/wiki/Orthogonal_group.
The unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.
The purpose of researching this paper is, as I have already stated in the summary, to gather as much information as possible and introduce it into the basic meaning of Lie groups. That everyone can benefit from this work at the beginning and find guidelines for further study of Lie groups. To get the most important information through this work without having to use a lot of literature like me and to get information in an easier way.
2. Groups, Subgroups, Definitions and Examples
Definition 1: Group is a set G together with a map
;
and an element
; such that the following conditions are fulfilled
1) An associative algebra is algebra A whose associative rule is associative:
for all
2) There exists an element
, such that for all
we have
. Such an element
is called an identity in G.
3) For every
there exists an element
such that
is called an inverse of a in G.
Definition 2: If in the group (G; ·) for all
is
;
Then we say that this structure is commutative (or Abel’s), so we can speak of a commutative group.
Example 1: Let (G, ·) be a group in which
holds for each element a. We claim that is then G a commutative group. Namely,
Example 2: If (V; +; ·) is a vector space, then (V; +) is an Abel group.
Definition 3: Let
be a one-membered set and define a binary operation on S by the formula
. Then ({e}; ·) is a group we call a trivial group.
Definition 4: Subgroup of G is a subset
such that
1)
;
2)
for all
and
;
3)
for every
.
Definition 5: [2] nonempty subset H of a group G is a subgroup of G if H is a group under the same operation as G. We use the notation
to mean that H is a subset of G, and H ≤ G to mean that H is a subgroup of G. For a group G with identity element e, {e} is a subgroup of G called the trivial subgroup. For any group G, G itself is a subgroup of G, called the improper subgroup. Any other subgroup of G besides the two above is called a nontrivial proper subgroup of G.
Definition 6: Let (G, ·) and (H, ∗) groups. Function
is a homomorphism if for all
is valid:
1)
2)
for all
Proof: a) If
,
and H is group, hitting the left with
,
, so
.
Definition 7: If
is a homomorphism. The kernel of f defined by
is also readily seen to be a subgroup of G.
Proof. We have to show that the kernel is non-empty and closed under products and inverses. Note that
. Thus Kerf is certainly non-empty. Now suppose that a and bare in the kernel, so that
.
.
Thus
and so the kernel is closed under products. Finally suppose that
. Then
. Thus the kernel is closed under inverses, and the kernel is a subgroup.
Example 3: Let
denote the set of positive real numbers. Then there is the function
is a homomorphism of a group
into a group
.
Example 4: Conjugation, i.e., function
, given by
, represents the automorphism of the algebra
Function
is a homomorphism
- Subjective group homomorphism is called an epimorphism
- Injective group homomorphism is called a monomorphism.
- Bijective group homomorphism is called an isomorphism.
- If
, we say that x is an endomorphism.
- If f is an isomorphism and an endomorphism, we say that x is an automorphism.
- Automorphism of G we mean an isomorphism of G onto itself.
Definition 9: The map
is called an isomorphism and G and H are said to be isomorphic if
1) f is a homomorphism.
2) f is a bijection.
- If G is a group and
; then the map
; is called left translation by x.
- If
; then
is called right translation by x
- If
; then
is called conjugation by x:
LIE GROUPS, DEFINITION AND EXAMPLES
A Lie group is a smooth (i.e., C 1) manifold G equipped with a group structure so that the maps
and the inversion map
are smooth.
In other words, the coordinates of the product must be differentiable functions of the coordinates of factors, and the coordinates of the inverse element must be differentiable functions of the coordinates of the element itself.
A Lie group over
is also called a complex Lie group and a Lie group over
is called a real Lie group. Any complex Lie group may be considered as a real Lie group of doubled dimension.
Example 1:
1)
together with addition+ and the neutral element 0 is a Lie group.
2)
is an open subset of R; hence a smooth manifold. Equipped with the ordinary scalar multiplication and the neutral element 1;
is a Lie group.
3)
together with addition + and the neutral element 0 is a Lie group.
4)
is an open subset of C; hence a smooth manifold. Equipped with the ordinary scalar multiplication and the neutral element 1;
is a Lie group.
Example 2: Let
Equipped with the group product given by
Then G is a Lie group.
[3] The General Linear Group GL(n, R)
Let n be a positive integer, and let
be the set of real n × n matrices. Equipped with entry wise addition and scalar multiplication
is a linear space, which in an obvious way may be identified with
. For
we denote by
the entry of A in the i-th row and the j-th column.
The maps
may be viewed as a system of (linear) coordinate functions on
. In terms of these coordinate functions, the determinant function
is given by
where
denotes the group of permutations of
and where sgn denotes the sign of a permutation. It follows from this formula that det is smooth.
The set
of invertible matrices in
equipped with the multiplication of matrices, is a group. As a set it is given by
Thus,
is the pre-image of the open subset
of R under det. As the latter function is continuous, it follows that
is an open subset of
. As such, it may be viewed as a smooth manifold of dimension
. In terms of the coordinate functions
the multiplication map
is given by
It follows that µ is smooth. Given
we denote by
the transpose of A. Moreover, for
n we denote by
the matrix obtained from A by deleting the i-th row and j-th column.
The co-matrix of A is defined by
Clearly, the map
is a polynomial, hence smooth map from
to itself. By Cramer’s rule the inversion
,
is given by
It follows that
is smooth, and we see that
is a Lie group.
Obviously
is an n2-dimensional noncompact Lie group, and it is not connected. In fact, it consists of exactly two connected components,
[4]: The Complex General Linear Group GL(n; C)
This calls for some explanation, since
is not a group of real matrices, as required by Definition:. A linear group is a closed subgroup of
. However, we can represent each complex matrix
by a real matrix
in the following way. If we “forget” scalar multiplication by non-reals, the complex vector space
becomes a real vector space RV of twice the dimension, with basis
,
,
,
,
,
,
:
Moreover each matrix
, i.e. each linear map
Defines a linear map
i.e. a matrix
.
Concretely, in passing from Z to RZ each entry
is replaced by the 2 × 2 matrix
The map
is injective; and it preserves the algebraic structure, i.e.
·
·
·
·
·
It follows in particular that RZ is invertible if and only if Z is; so R restricts to a map
Whenever we speak of
, or more generally of any group G of complex matrices, as a linear group, it is understood that we refer to the image RG of G under this injection R.
The matrix
belongs to
if is built out of 2 × 2 matrices of the form
This can be expressed more neatly as follows. Let
Since any scalar multiple of the identity commutes with all matrices,
Applying the operator R,
Conversely, if
then it is readily verified that X is of the required form.
Thus
and in particular
[3]: The Special Linear Group S(n, R) = SL(V)
Let V be a finite dimensional real linear space. We define the special linear group
Note that det is a group homomorphism from
to
. Moreover,
is the kernel of det. In particular,
is a subgroup of
. We will show that
is a sub-manifold of
of co-dimension 1. Suffices to do this at the element = IV.
Since
is an open subset of the linear space
its tangent space
may be identified with
. The determinant function is smooth from G to R hence its tangent map is a linear map from
to R. Tangent map is the trace
. Clearly tr is a surjective linear map. This implies that det is submersive at I. That
is a smooth co-dimension 1 sub-manifold at I:
[4]. The Complex Special Linear Group SLn,
Note that the determinant here must be computed in
, not in
.
Thus
Although
[5] Unitary and Orthogonal Groups
An
complex matrix A is said to be unitary if the column vectors of A are orthonormal, that is, if
We may rewrite as
where
is the Kronecker delta equal to 1 if
and equal to zero if
.
Here A is the adjoint of A, defined by
Equation says that
; thus, we see that A is unitary if and only if
. In particular, every unitary matrix is invertible.
The adjoint operation on matrices satisfies
. from this, we can see that if A and B are unitary, then
showing that AB is also unitary. Furthermore, since
, we see that
, which shows that
. Thus, if A is unitary, we have
showing that
is again unitary.
Thus, the collection of unitary matrices is a subgroup of
. We call this group the unitary group and we denote it by
. We may also define the special unitary group
, the subgroup of
consisting of unitary matrices with determinant 1. It is easy to check that both
and
are closed subgroups of
and thus matrix Lie groups.
Meanwhile, let
denote the standard inner product on
, given by
(Note that we put the conjugate on the first factor in the inner product.)
We have
for all
. Thus
from which we can see that if A is unitary, then A preserves the inner product on
, that is,
for all x and y. Conversely, if A preserves the inner product, we must have
for all
. It is not hard to see that this condition holds only if
. Thus, an equivalent characterization of unitarity is that A is unitary if and only if A preserves the standard inner product on
.
Finally, for any matrix A, we have that
. Thus, if A is unitary, we have
Hence, for all unitary matrices A, we have
.
In a similar fashion, an
real matrix A is said to be orthogonal if the column vectors of A are orthonormal. As in the unitary case, we may give equivalent versions of this condition. The only difference is that if A is real, A is the same as the transpose
of A, given by
Thus, A is orthogonal if and only if
, and this holds if and only if A preserves the inner product on
. Since
, if A is orthogonal, we have
so that
. The collection of all orthogonal matrices forms a closed subgroup of
. which we call the orthogonal group and denote by
.
The set of
orthogonal matrices with determinant one is the special orthogonal group, denoted
. Geometrically, elements of
. are rotations, while the elements of
. are either rotations or combinations of rotations and reflections. Consider now the bilinear form
on
defined by
This form is not an inner product because, for example, it is symmetric rather than conjugate symmetric. The set of all
complex matrices A which preserve this form (i.e., such that
for all
) is the complex orthogonal group
, and it is a subgroup of
. Since there are no conjugates in the definition of the form
, we have
for all
where on the right-hand side of the above relation, we have
rather than
. Repeating the arguments for the case of
, but now allowing complex entries in our matrices, we find that an
complex matrix A is in
, if and only if
, that
, is a matrix Lie group, and that
for all A in
. Note that
is not the same as the unitary group
.
The group
is defined to be the set of all A in
with
and it is also a matrix Lie group.
[5] Symplectic Groups
Consider the skew-symmetric bilinear form B on
defined as follows
The set of all
real matrices A which preserve
(i.e., such that
for all
) is the real symplectic group
, and it is a closed subgroup of
. (Some authors refer to the group we have just defined as
rather than
If is the
matrix
Then
From this, it is not hard to show that a
real matrix A belongs to
if and only if
Taking the determinant of this identity gives
, i.e.
. This shows that
, for all
. In fact,
for all
, although this is not obvious. One can define a bilinear form
on
by the same formula as in
(with no conjugates).
Over C, we have the relation
where
is the complex bilinear form in
. The set of
complex matrices which preserve this form is the complex symplectic group
. A
complex matrix A is in
if and only if
holds. (Note: This condition involves
not A.) Again, we can easily show that each
. Satisfies
and, again, it is actually the case that
. Finally, we have the compact symplectic group
defined as
[6]: The groups
and
The groups
and
under matrix multiplication are isomorphic to
and
, respectively, and so we view them as matrix Lie groups. The group
of complex numbers with absolute value one is isomorphic to U(1) and so we also view it as a matrix Lie group. The group
under vector addition is isomorphic to the group of diagonal real matrices with positive diagonal entries, via the map
One easily checks that this is a matrix Lie group and thus we view
as a matrix Lie group as well.
Definition 2. [7] A subgroup H of a Lie group G is called a Lie subgroup if it is a Lie group (with respect to the induced group operation), and the inclusion map
is a smooth immersion (and therefore a Lie group homomorphism).
Example 8. [7] Consider
. Then
and
are Lie subgroups. Moreover, for any co-prime pair of integers (p; q),
is a Lie subgroup of
. These are submanifolds as well. However, there are also many Lie subgroups of
which are not submanifolds. In fact, for any irrational number α,
is a Lie subgroup of
. But
, so they are not submanifolds.
Definition 3. A Lie subgroup H of G is said to be a closed Lie subgroup if H is both a Lie subgroup and also a submanifold of G.
Lemma 1 [7]. Suppose G is a Lie group, H is a subgroup of G which is a submanifold as well. Then H is closed in the sense of topology.
Proof. Since H is a submanifold of G, it is locally closed everywhere. In particular, one can find an open neighborhood U of e in G such that
. Now take any
. Since hU is an open neighborhood of h in G,
. Let
, then
. On the other hand, since
, there is a sequence
in H converging to h. It follows that the sequence
converges to
.
In other words,
. So
, i.e.
. Therefore, H is closed.
3. Homomorphisms of Lie Groups
Lie group homomorphism
Let G and H be Lie groups. A map
is called a Lie group homomorphism if
1)
is a group homomorphism, and
2)
is continuous.
Lie group isomorphism
Let G and H be Lie groups. A map
is called a Lie group isomorphism if
1)
is one-to-one and onto, and
2) the inverse map is
continuous.
Examples 1 [6]:
1) The map
given by
is a Lie group homomorphism.
2) The map
given by
is a Lie group isomorphism (you should check that this map is well-defined and is indeed an isomorphism).
1) Composing the previous two examples gives the Lie algebra homomorphism
defined by
2) The determinant is a Lie group homomorphism
.
4. Invariant Vector Fields and the Exponential Map
Definition 1. A vector field
is left-invariant if
for every
, and right-invariant if
for every
. A vector field is called bi-invariant if it is both left- and right-invariant.
Theorem 1 [8]. The map
(where 1 is the identity element of the group) defines an isomorphism of the vector space of left-invariant vector fields on G with the vector space
, and similarly for right-invariant vector spaces. Proof. It suffices to prove that every
can be uniquely extended to a left-invariant vector field on G. Let us define the extension by
. Then one easily sees that the so-defined vector field is left-invariant, and
. This proves the existence of an extension; uniqueness is obvious.
Definition 2. A smooth homomorphism
is called a one parameter group subgroup of G.
Definition 3. Let G be a real or complex Lie group. Then the exponential map
is defined by
where
is the one-parameter subgroup with tangent vector at 1 equal to x.
Example 3 [3]. We return to the example of the group
with V a finite dimensional real linear space. Its neutral element e equals
. Since
is open in
, we have
. If
then
is the restriction of the linear map
;
to
hence
. Hence, the integral curve
satisfies the equation:
Since
is a solution to this equation with the same initial value, we must have that
. Thus in this case exp is the ordinary exponential map
,
.
Proposition 1 [9]. The exponential map
satisfies:
1) For each
,
is a one parameter group with
.
2) The integral curve c of the left invariant vector field
with
is
.
3) exp is smooth with
.
4) If
is a Lie group homomorphism, then
for
.
5) If
is a Lie subgroup then
Proof: First observe that
is an integral curve of X through e since
Thus
, since, for fixed t, both are integral curves of tX through e. To see this for the right hand side, observe that in general if γ(s) is an integral curve of a vector field X, then γ(ts) is an integral curve of tX. Hence
, which implies (a). Since
takes integral curves to integral curves, (b) follows as well.
To see that exp is smooth, define a vector field Z on
by
. Z is clearly smooth and by part (b), its flow is
. Thus
is smooth in X and hence exp is smooth as well. Finally,
, which proves the second claim in (c).
To prove (d), observe that a homomorphism takes one parameter groups to one parameter groups. Thus
is a one parameter group with
and hence
, which proves our claim by setting t = 1.
Part (e) follows easily by applying (d) to the inclusion of H in G.
Example 4.
and
, the set of invertible matrices, are Lie groups. For these groups we claim that
, which explains the name exponential map. Indeed, from the power series definition of
it easily follows that
, i.e.
is a one parameter group. Furthermore
and hence
.
Lemma 1 [3]: Let
be a homomorphism of Lie groups. Then the following diagram commutes
Proof: Let
. Then
is a one-parameter subgroup of H: Differentiating at
we obtain
. Now apply the above lemma to conclude that
. The result follows by specializing to
.
5. The Lie Algebra of a Lie Group
For
consider the conjugation map
. Since
is a homomorphism
is a Lie algebra homomorphism.
The map
is called the adjoint representation of G in
.
For
let
be defined by
. The Jacobi identity is equivalent to saying that
i.e.
is defined by
We note that, by the chain rule, for all
;
Proposition [5]. If
is a Lie algebra, then
that is,
is a Lie algebra homomorphism.
Proof. Observe that
whereas
Thus, we want to show that
which is equivalent to the Jacobi identity
Lemma 1: The adjoint representation satisfies:
1)
or simply
2)
Proof: For part a) we see that for any
where
is the Lie derivative. In the last passage, we used the definition of Lie derivative, and the fact that
is the flow of X.
We may apply Lemma 1 with
and
. Since
whereas
is given by
; we see that the following diagram commutes:
Example 1. [3] Let V be finite dimensional real linear space. Then for
the linear map
is given by
: Substituting
and differentiating the resulting expression with respect to t at
we obtain:
Hence in this case.
is the commutator bracket of X and Y.
Definition 3. For
we define the Lie bracket
by
Example 2. Let X and Y be
matrices. Show by induction that
where
Now, show by direct computation that
Lemma 1. [3] Let
be a homomorphism of Lie groups. Then the following diagram commutes:
Proof: One readily verifies that
. Taking the tangent map of both sides of this equation at e, we obtain that the following diagram commutes:
Differentiating once more at x D e; in the direction of
; we obtain that the following diagram commutes:
We now agree to write
.
6. The Future Perspective of This Paper
The future perspective of this paper is that since this is a work of a general nature, it is necessary to extract the most important things from it. It is possible to do a paper for each of the titles in the paper. Each of the titles creates an opportunity for research because from each title there are many opportunities to explore and write. It is my future to work on comparing many facts and the application of leftist groups in everyday life. To give a glimpse of a better tomorrow of this beautiful science called both left groups and left algebras and their close connection to other mathematical disciplines primarily thinking of linear algebra, geometry, analysis and topology. Comparing all these disciplines with the left groups, we see a close connection and the need to apply and use them in the right way.