1. Introduction
A root system in mathematics is a configuration of vectors in Euclidean space that satisfies certain geometric properties. The concept is fundamental in the theory of Lie algebras and Lie groups, especially in the theory of classification and representation of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras became important in many parts of mathematics during the twentieth century, the apparently special nature of root systems is inconsistent with the number of areas in which they are applied.
Definition 1.1. A root system
is a finite-dimensional real vector space V wich an inner product (i.e. a Euclidean vector space), such that the following properties hold:
a) The vectors in R span V.
b) If α is in R and
, then
is in R only if
, then R is called a reduced root system.
c) For any two roots
, the number
is integer.
The dimension of V is called the rank of the root system and the elements of R are called roots.
d) Let
be defined by
for
. Another calculation also shows that
preserves the inner product
, i.e.,
for
, that is,
is in the orthogonal group
. Evidently,
:
Theorem 1.2. Suppose that α and β are linearly independent elements. Then
1.
.
2. If
, then
.
3. If
and
, then
and
4. If
and
, then
and
5. If
and
, the
. If
and
, then
.
Proof. 1. Since α and β are linearly independent, the Cauchy-Schwartz Inequality implies that
.
Hence
Since
and
are integers, we must have
2. Obvious. If
, then
.
3. If
and
,
So
.
. we must have
, whence
. Thus
4. If
and
, then
So
. By Part (1),
. Since
is the smaller integer factor, we must have
, whence
. Thus
5. Suppose that
and
. Then by Part (4),
, so
. On the other hand, if
and
, then by Part (4),
, whence
.
The novelty of this work is based on the connection of root systems with Lie algebra. In addition, the importance of Dynkin diagrams and Ways chamebers with the rook system and Lie algebra is shown. Modern pictures and graphs were used with the help of modern tools to more closely convey the appearance and quality of the work.
2. Basic Theory of Root Systems
Theory 2.1. [1] Let V be a finite-dimensional vector space over R equipt with an inner product
. The Cauchy-Schwartz inequality asserts that
for
. It follows that if
are nonzero, then
If
are nonzero, then we define the angle between u and v to be the unique number
such that
The inner product measures the angle between two vectors, though it is a bit more complicated in that the lengths of x and y are also involved. The term “angle” does make sense geometrically.
Proposition 2.2. [2] Suppose that α and β are linearly independent roots and
. Let θ be the angle between α and β. Then we have the following Table 1 & Figure 1.
Proof. Here
, so
. Now
, or 3 and
. Moreover,
has the same sign as
. This gives us Table 1 below.
Table 1.
, or 3 and
.
Figure 1. The allowed angles and length ratios, for the case of an acute angle.
Lemma 2.3. If
, then
.
Proof If 1, 2 or 3 is written as a product of two positive integers, then one of the factors is 1. Up to swapping α and β we can assume that
. The reflection in α sends β to
, thus
.
Definition 2.4. Suppose
and
are root systems. Consider the vector space,
with the natural inner product determined by the inner products on
and
. Then
is a root system in
, called the direct sum of R and Q.
Definition 2.5. A root system
is called reducible if there exists an orthogonal decomposition
with
and
such that every element of R is either in
or in
. If no such decomposition exists,
is called irreducible.
Example 2.6. [3] The following Figure 2 shows the root systems of rank 1 and 2. All of them are indecomposable (except
), and reduced (except
and
).
Proposition 2.7. Every root system of the second rank is isomorphic to one of the systems in Figure 2.
Proof: Suppose that
; therefore, let
be the root system. Let be the smallest angle occurring between any two vectors in R. Since the elements of R span R2, we can find two linearly independent vectors α and β in R. If the angle between α and β is greater than
, then the angle between α and -β is less than
therefore, the minimum angle is at most
.
3. Root Systems For Classical Complex Lie Algebras
Definition 3.1. A root
is said to be simple if α is positive and α is not the sum of two positive roots. The collection Γ of all simple roots is called a simple system of roots.
Lemma 3.2. Let
be a simple system of roots. Then every positive root δ can be written as
, where each initial partial sum
(
) is a root.
Proof. For every positive root
, the height of δ is defined to be the positive number ht
We prove this lemma by induction on ht δ. If
, then δ is simple and there is nothing to prove. So assume that m > and that the lemma’s conclusion holds for all positive roots of height < m. Now suppose that δ is a positive root of height m. Now apply the induction hypothesis to the root
, which has height
. Then
, where each initial partial sum is a root. Then
. Thus δ satisfies the conclusion of the lemma, completing the induction step as well as the proof.
Definition 3.3. The root system Λ is decomposable if Λ is a union
with
,
, and
.
Definition 3.4. If Γ is a simple system of roots in Λ, we say that is Γ decomposable if Γ is a union
, with
,
, and
.
Lemma 3.4. Let Γ be a simple system of roots in Λ. Then Λ is decomposable if and only if Γ is decomposable.
Proof. Suppose that Γ is decomposable, with
For
, let
. Then neither
nor
can be empty. For if, say
, then
which implies that
Since Γ is a basis of E, we conclude that
, and so
, contradiction.
Conversely, suppose that Γ is decomposable, with
. We arrange the elements of Γ so that
and
. Now let
. We claim that δ is a linear combination of elements of
or δ is a linear combination of elements of
. To prove this claim, we may assume that δ is positive. Now suppose, to the contrary, that δ is a linear combination
where both sums on the right are nonzero. Without loss of generality, we can assume that
. Let s be the smallest integer such that
. Then
is a root.
Now consider the root
. This root equals
which is not a linear combination of simple roots with nonnegative integer coefficients, a contradiction. This proves the claim.
Using the claim, we now let
be the set of roots which are linear combinations of elements of
, and let
be the set of roots which are linear combinations of elements of
. Then
,
,
, and
. Thus Λ is decomposable.
Definition 3.5. Let Γ be a simple system of roots in Λ. Then Λ is decomposable if and only if Γ is decomposable.
Example 3.6. [2] Let E be a two-dimensional inner product space. We will show that, up to isometry, there are only three possible indecomposable simple systems of roots Γ on E. Suppose that
. Then
, since Γ is indecomposable. We may assume that
.
and
(Figures 3-5).
Figure 3.
and
.
Figure 4.
and
.
Figure 5.
and
.
Let
. be a simple system of roots in Λ. We introduce a partial ordering
on Λ as follows: if
, then
if and only if
where each
and at least one
is positive. It is clear that
is indeed a partial order on Λ. Of course,
depends on the choice of Λ. Recall that the simple system Λ was obtained via a lexicographic order < on E. Since each simple root αi is a positive root under <, it is clear that if α and β are roots such that
, then
. The converse is not true, as there are vectors in Λ which are not comparable under <.
4. Weyl Chambers
Definition 4.1. Let R be a root system in V. The hyperplanes
subdivide F into finitely many polyhedral convex cones. We recall that each root
. The elements of the set
Also, recall that a vector
is regular with respect to R if and only if
Evidently,
is an open subset of V. A path component of the space
is called a Weyl chamber of V with respect to R.
If C is a Weyl chamber, then
is also a Weyl chamber. It is called the Weyl chamber opposite to C. A hyperplane
is called a wall of the Weyl chamber C if
and
contains a nonempty subset open in P.
A subsystem Π of a root system Λ is called a system of simple roots (ora base) of the system n if Π is linearly independent and each
can be represented in the form
where
, are integers, which are simultaneously either nonpositive or nonnegative. In the first case is β said to be positive (β > 0), in the second negative (β < 0) with respect to Π.
Lemma 4.2. [3] For any Weyl chamber C the system Π(C) is a system of simple roots. The roots that are positive (negative) with respect to Pi(C) coincide with C-positive (respectively, C-negative) roots. The correspondence
between the Weyl chambers and systems of simple roots is bijective. For any Weyl chamber C, we have
The walls of the Weyl chamber C are the hyperplanes
where
.
Proof. The closure of C consists of C and points
with
such that there exists a sequence
of elements of C such that
as
Let x be an element of C the this second type. Assume that there exists
such that
. Since
as
, there exists a positive integer n such that
. This is a contradiction. It follows that
is contained in
. Let x be in
. we need to prove that
. Let
. Consider the sequence
Evidently this sequence converges to x and is contained in C. It follows that x is in
. This proves the first assertion of the lemma. For the second assertion, let
. If
, then v is by definition in some Weyl chamber. Assume that
. Then
, Define
The function p is a non-zero polynomial function on V, and the set of zeros of p is exactly
. Thus,
. Since p is a non-zero polynomial function on V, p cannot vanish on an open set. Hence, for each positive integer n, there exists vn such that
and
. The sequence
converges to x and is contained in
; in particular every element of the sequence is contained in some Weyl chamber. Since the number of Weyl chambers of V with respect to R is finite. We have
for all
and positive integers k. Taking limits, we find that
for all
, so that
.
Example.4.3. [3] Using the lexicographic order with respect to the basis composed of the weights
, one can easily construct systems of simple roots
, in the root systems
, of the classical Lie algebras g,
of
, where
The corresponding Weyl chamber consists of the set of diagonal matrices
such that
,
where
,
where
,
where
,
Lemma 4.4. In this situation, if α and β are not orthogonal then
,
and
Proof: We know that
Taking the product,
but α and β are neither proportional nor perpendicular, so
where k = 1; 2, or 3. Since
, the first term in the first equation is the smaller integer, hence
Straightforward manipuulations of this imply what we want.
Corollary 4.5. Suppose α,β are distinct simple roots and
. Then
with
, or vice versa.
5. Cartan Matrices And Dynkin Diagrams
Definition 5.1. A system
is said to be admissible if
is a nonpositive integer for all
. The integer square matrix
is called the matrix of the system Φ. Let
and let
be the angle between the vectors
and
(
), implies that for an admissible system Φ the numbers
and the angles
can assume only the
following values:
;
, where
, respectively.
Definition 5.2. The Dynkin diagram of an admissible system of vectors is the graph described above in which the edge joining the vertices numbered by i and j (
,
) is of multiplicity
. If
, then the corresponding edge is oriented by an arrow pointing from the j-th vertex towards the i-th one.
Theorem 5.3. The Dynkin diagrams of the classical simple Lie algebras g are of the following list.
The An root lattice—that is, the lattice generated by the An roots—is most easily described as the set of integer vectors in
whose components sum to zero (Figure 6).
Example: The A3 root lattice is known to crystallographers as a face-centered cubic lattice.
Simple roots in A3
The A3 root system (as well as other third-order root systems) can be modeled in the Zometool Construction set (Figures 7-9).
Root system-WikipediaModel of the root system in the Zometool system.Brian C. Hall “Lie Groups, Lie Algebras, and Representations”—Fig. 8.16, 229.
Figure 7. The roots in A3 make up the vertices of a cuboctahedron.
Brian C. Hall “Lie Groups, Lie Algebras, and Representations”—Fig. 8.17, 229.
Figure 8. The roots in A3 lie at the midpoints of the edges of a cube.
The Bn root lattice—that is, the lattice generated by the Bn roots—consists of all integer vectors.
Example:
B1 is isomorphic to A1 via scaling by
, and is therefore not a distinct root system (Figure 10).
Simple roots in B4
Brian C. Hall “Lie Groups, Lie Algebras, and Representations”—Fig. 8.18, 230.
Figure 10. The B3 root system, with the elements of the base in dark gray.
The Cn root lattice—that is, the lattice generated by the Cn roots—consists of all integer vectors whose components sum to an even integer (Figure 11).
Example:
C2 is isomorphic to B2 via scaling by
and a 45 degree rotation, and is therefore not a distinct root system (Figure 12, Figure 13).
Simple roots in C4
Brian C. Hall “Lie Groups, Lie Algebras, and Representations”—Fig. 8.20, 229.
Figure 12. Root system C3 with the elements of the base in dark gray.
Brian C. Hall “Lie Groups, Lie Algebras, and Representations”—Fig. 8.31, 230.
Figure 13. The C3 root system consists of the vertices of an octahedron, together with the midpoints of the edges of the octahedron.
The Dn root lattice—that is, the lattice generated by the Dn roots—consists of all integer vectors whose components sum to an even integer. This is the same as the Cn root lattice (Figure 14).
Example:
D3 coincides with A3, and is therefore not a distinct root system. The 12 D3 root vectors are expressed as the vertices of, a lower symmetry construction of the cuboctahedron.
D4 has additional symmetry called triality. The 24 D4 root vectors are expressed as the vertices of, a lower symmetry construction of the 24-cell. 72 vertices (Figure 15).
Simple roots in D4
72 vertices of 122 represent the root vectors of E6 (Figure 16, Figure 17, Figure 18 & Figure 19).
126 vertices of 231 represent the root vectors of E7.
240 vertices of 421 represent the root vectors of E8 (Figure 20, Figure 21).
Root system—Wikipedia.
Figure 21. E8 root system.
The F4 root lattice—that is, the lattice generated by the F4 root system is the set of points in R4 such that either all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and halfintegers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions (Figure 22, Figure 23).
Simple roots in F4
The G2 root lattice—that is, the lattice generated by the G2 roots—is the same as the A2 root lattice (Figure 24).
Simple roots in G2
The root system G2 has 12 roots, which form the vertices of a hexagram. One choice of simple roots is: (
) where
for
.
Example 5.4. [4] The extended Dynkin diagrams of simple classical Lie algebras are of the following form (each diagram contains n+1 vertices, the right column lists the standard notation for each of the diagrams) (Figure 25):
Lemma 5.5. A Coxeter-Dynkin graph is a tree (Figure 26).
Proof. Suppose, to the contrary, that there are circuits. Let
be the vertices of a minimal circuit.
Since the circuit is minimal, no root
is connected to a root
in the circuit unless
or
. Suppose now that
and
are consecutive roots in the circuit. We claim that
To show this, we may assume that
. Then obviously,
We have,
Adding the left hand sides of the last two relations above, we obtain inequality
. Thus, in particular,
for all
, where the index
is counted modulo n. Adding these inequalities, we obtain
On the other hand,
by our remark at the beginning of the proof about adjacent vertices. Inequalities
and
, imply that
. But this is a contradiction since the
are linearly independent.
Lemma 5.6. In a Dynkin diagram, suppose that roots γ and δ are joined by a simple edge. Then the configuration resulting from the deletion of γ and δ and replacement by the single root γ+δ, and then joining γ+δ to all roots connected to γ or δ by the same types of edges as γ or δ is also a Dynkin diagram.
Proof. Note first that since γ and δ are connected, we have
and thus γ + δ is a root. Moreover, since
, we have
and
. Hence
.
Let S be the collection of roots β in the Dynkin diagram such that
,
, and β is connected to γ or δ.
So let
. Without loss of generality, we can assume that β is connected to γ. Then
, and so
.
Moreover,
Hence
This shows that the number of bonds in the edge joining β and γ + δ is the same as the number of bonds in the edge joining β and γ.
Finally, since
, the direction of the edge joining β and γ + δ is the same as the direction of the edge joining β and γ (Figure 27).
Example
6. The Future Perspective of This Paper
The future of this work is related to dealing with Lie algebras. The beauty of this science is learned every day by doing as much research and work on it as possible. The close connection with other mathematical disciplines creates abundant opportunities for further research and dealing with what our work is in the future. The primary goal of the paper was to be useful to anyone studying Lie algebra. In addition, this paper shows the influence of Dynkin diagrams on the root system, and the beauty of their diagram, as a close connection with Ways chambers.