1 Introduction
Back in the 19th century, a mathematician named Sophus Lie introduced the subject of studying continuous symmetries of geometric objects and their derivatives to the mathematical world. These topics are now called Lie groups and Lie algebras respectively.
Since then, these topics have been far reaching in their capabilities; finding a role in many areas such as geometric objects, matrices, and symmetry to name a few. In addition to this, Lie algebras have also found use in several fields of study in physics such as string theory and quantum mechanics.
However, since the topic is still relatively new and is usually taught isolated from other mathematic classes, the capabilities of this subject are still unknown. In this paper, some of the basic elements and associations of Lie algebras will be discussed to give a brief background on this diverse topic.
2 Basic Properties
A Lie algebra is a vector space over a field that multiplies in such a way that a and b hold true for elements X,Y,Z and where [ ] represents multiplication in the vector space. See [1] pg 3, [4] pg 3, and [6] pg 25.
a. [X, X] = 0
b. [X[Y, Z]]+[Y[Z, X]]+[Z[X, Y]] = 0 which can also be written as:
[X[Y, Z]] + [Y [Z, X]] + [Z[X, Y]] = 0
adX.[Y, Z] = [adX.Y, Z] + [Y, adX.Z]
ad[X, Y]=adXadY − adYadX
These properties can be broken down by name. Suppose L is a Lie algebra and F is a field.
a. Bilinear Product
LxL = L
b. Antisymmetry or Skew Symmetry
[X,Y] = − [Y, X] for X,Y ∈ L
c. Linearity
[λX, Y ] = λ[X, Y ], [X + Y, Z] = [X, Z] + [Y, Z] for X,Y,Z ∈ L and λ ∈ F
d. Jacobi Identity
[X[Y, Z]] + [Y [Z, X]] + [Z[X, Y ]] = 0 for X, Y, Z ∈ L
Back in the 19th century, a mathematician named Sophus Lie introduced the subject of studying continuous symmetries of geometric objects and their derivatives to the mathematical world. These topics are now called Lie groups and Lie algebras respectively.
Since then, these topics have been far reaching in their capabilities; finding a role in many areas such as geometric objects, matrices, and symmetry to name a few. In addition to this, Lie algebras have also found use in several fields of study in physics such as string theory and quantum mechanics.
However, since the topic is still relatively new and is usually taught isolated from other mathematic classes, the capabilities of this subject are still unknown. In this paper, some of the basic elements and associations of Lie algebras will be discussed to give a brief background on this diverse topic.
2 Basic Properties
A Lie algebra is a vector space over a field that multiplies in such a way that a and b hold true for elements X,Y,Z and where [ ] represents multiplication in the vector space. See [1] pg 3, [4] pg 3, and [6] pg 25.
a. [X, X] = 0
b. [X[Y, Z]]+[Y[Z, X]]+[Z[X, Y]] = 0 which can also be written as:
[X[Y, Z]] + [Y [Z, X]] + [Z[X, Y]] = 0
adX.[Y, Z] = [adX.Y, Z] + [Y, adX.Z]
ad[X, Y]=adXadY − adYadX
These properties can be broken down by name. Suppose L is a Lie algebra and F is a field.
a. Bilinear Product
LxL = L
b. Antisymmetry or Skew Symmetry
[X,Y] = − [Y, X] for X,Y ∈ L
c. Linearity
[λX, Y ] = λ[X, Y ], [X + Y, Z] = [X, Z] + [Y, Z] for X,Y,Z ∈ L and λ ∈ F
d. Jacobi Identity
[X[Y, Z]] + [Y [Z, X]] + [Z[X, Y ]] = 0 for X, Y, Z ∈ L
2.1 Importance
Lie algebras are heavily used in particle physics and in string theory. They are also applied in differential geometry, Lie groups, as well as in Hamilton and Quantum mechanics. While these are the main uses for Lie algebras, they can be applied to most fields of mathematics. [7]
3 Lie Groups
Before Lie algebras can be understood, Lie groups should be explained as Lie algebras originated as a way to study Lie groups. A Lie group is simply a topo- logical group that is a manifold. This means that a Lie group is merely a group that differential calculus may be applied to.
3.1 Groups
To be a group, certain properties must be met. [9], pg 172.
a. ForA,B ∈ G, AB ∈ G.
b. For A,B,C ∈ G, A(BC) = (AB)C
c. There is an identity I such that IA = AI = A for every A ∈ G.
d. For every A in G, there exists B = A−1 such that AA−1 = A−1A = I
To satisfy the requirements to be specifically a Lie group, the operations must also be differentiable. A morphism of Lie groups is a smooth map which follows f(gh) = f(g)f(h) and f(1) = 1
Remark: To be smooth means to have a derivative for every order in the domain.
Lie algebras are heavily used in particle physics and in string theory. They are also applied in differential geometry, Lie groups, as well as in Hamilton and Quantum mechanics. While these are the main uses for Lie algebras, they can be applied to most fields of mathematics. [7]
3 Lie Groups
Before Lie algebras can be understood, Lie groups should be explained as Lie algebras originated as a way to study Lie groups. A Lie group is simply a topo- logical group that is a manifold. This means that a Lie group is merely a group that differential calculus may be applied to.
3.1 Groups
To be a group, certain properties must be met. [9], pg 172.
a. ForA,B ∈ G, AB ∈ G.
b. For A,B,C ∈ G, A(BC) = (AB)C
c. There is an identity I such that IA = AI = A for every A ∈ G.
d. For every A in G, there exists B = A−1 such that AA−1 = A−1A = I
To satisfy the requirements to be specifically a Lie group, the operations must also be differentiable. A morphism of Lie groups is a smooth map which follows f(gh) = f(g)f(h) and f(1) = 1
Remark: To be smooth means to have a derivative for every order in the domain.
For example, SL(2, R) =
for k, l, m, n ∈ R and kn − lm = 1
is a simple real Lie group.
is a simple real Lie group.
Non zero real numbers with multiplication are also Lie groups.
On the other hand, an uncountable direct sum of a finite cyclic group would not be a Lie group.
On the other hand, an uncountable direct sum of a finite cyclic group would not be a Lie group.
3.2 Connections to Lie Algebras
A Lie algebra is the tangent space of a Lie group at the unit element. [1], pg 16. Put another way, a Lie algebra is a logarithm of a Lie group and a Lie group is an exponential of a Lie algebra. Since a Lie algebra is linear, it is often easier to manipulate than its Lie group.
Suppose L is a Lie algebra of Lie group, G. Then the structure of G near the identity is determined by L.
Example
Let G be a Lie group and let A, B ∈ gl(n) where gl(n) is the subspace of real nxn matrices.
Then eA+B = lim k → ∞(eA/keB/k)k.
Lie(G) = A ∈ gl(n):etA ∈ G, ∀t ∈ R.
Let A, B ∈ Lie(G).
Then et(A+B) = lim k → ∞(etA/ketB/k)k ∈ G.
Therefore, A + B ∈ Lie(G).
If A ∈ Lie(G) then αA ∈ Lie(G).
Therefore Lie(G) is a vector subspace of gl(n) and closed under scalar multiplication. [8], pg 4.
QED
4 Jacobi Identity
The Jacobi Identity is a large part of what makes a Lie algebra just that. How- ever, in the 20th century is was discovered that this aspect of a Lie algebra has uses separate from Lie Theory. [3]
The Jacobi has been applied to many things including:
The Double Dualization Functor
Tangent-Vector-Valued Differential Forms
Schwartz Distributions
Each of these subject fields are very complicated and would take up too much space to explain here. Basic Theorems regarding their relation to the Jacobi Idenity can be found in [3].
4.1 Example 1
Prove (A × B) × C + (B × C) × A + (C × A) × B = 0
Keep in mind that the cross product is not associative meaning that A × (B × C) ≠= (A × B) × C
A Lie algebra is the tangent space of a Lie group at the unit element. [1], pg 16. Put another way, a Lie algebra is a logarithm of a Lie group and a Lie group is an exponential of a Lie algebra. Since a Lie algebra is linear, it is often easier to manipulate than its Lie group.
Suppose L is a Lie algebra of Lie group, G. Then the structure of G near the identity is determined by L.
Example
Let G be a Lie group and let A, B ∈ gl(n) where gl(n) is the subspace of real nxn matrices.
Then eA+B = lim k → ∞(eA/keB/k)k.
Lie(G) = A ∈ gl(n):etA ∈ G, ∀t ∈ R.
Let A, B ∈ Lie(G).
Then et(A+B) = lim k → ∞(etA/ketB/k)k ∈ G.
Therefore, A + B ∈ Lie(G).
If A ∈ Lie(G) then αA ∈ Lie(G).
Therefore Lie(G) is a vector subspace of gl(n) and closed under scalar multiplication. [8], pg 4.
QED
4 Jacobi Identity
The Jacobi Identity is a large part of what makes a Lie algebra just that. How- ever, in the 20th century is was discovered that this aspect of a Lie algebra has uses separate from Lie Theory. [3]
The Jacobi has been applied to many things including:
The Double Dualization Functor
Tangent-Vector-Valued Differential Forms
Schwartz Distributions
Each of these subject fields are very complicated and would take up too much space to explain here. Basic Theorems regarding their relation to the Jacobi Idenity can be found in [3].
4.1 Example 1
Prove (A × B) × C + (B × C) × A + (C × A) × B = 0
Keep in mind that the cross product is not associative meaning that A × (B × C) ≠= (A × B) × C
Proof
(A × B) × C + (B × C) × A + (C × A) × B = (AC)B − (BC)A + (BA)C − (CA)B + (CB)A − (AB)C = 0
QED
4.2 Example 2
Let L be an algebra over F.
[XY ] = XY − Y X defines L to be a Lie algebra because the Jacobi Identity holds.
The Jacobi Identity may also be proved using the Poisson bracket. The Poisson bracket is very important in dynamics and when the bracket is not met because the Jacobi Identity is not fulfilled, those systems (coined almost- Poisson) become associated with dissipation and waste.
5 Basic Theorems
To gain a better understanding of the complex relations that Lie algebras may find themselves in, a few examples of basic theorems is shown below. [2]
Theorem 1
Let L be a Lie algebra in field F. Let I be a subideal of L.
a. If I is a nilregular ideal of L then N(I) ⊆ N(L)
b. If I is a nilregular subideal of L and every subideal of L containing I is nil- regular, then N(I) ⊆ N(L)
Corollary
Let L be a Lie algebra over a field F. Then every minimum ideal of L is abelian, simple, or irregular.
Theorem 2
Let L be a Lie algebra.
a. If L is a solvable primitive Lie algebra then all core-free maximal subalgebras are conjugate.
(A × B) × C + (B × C) × A + (C × A) × B = (AC)B − (BC)A + (BA)C − (CA)B + (CB)A − (AB)C = 0
QED
4.2 Example 2
Let L be an algebra over F.
[XY ] = XY − Y X defines L to be a Lie algebra because the Jacobi Identity holds.
The Jacobi Identity may also be proved using the Poisson bracket. The Poisson bracket is very important in dynamics and when the bracket is not met because the Jacobi Identity is not fulfilled, those systems (coined almost- Poisson) become associated with dissipation and waste.
5 Basic Theorems
To gain a better understanding of the complex relations that Lie algebras may find themselves in, a few examples of basic theorems is shown below. [2]
Theorem 1
Let L be a Lie algebra in field F. Let I be a subideal of L.
a. If I is a nilregular ideal of L then N(I) ⊆ N(L)
b. If I is a nilregular subideal of L and every subideal of L containing I is nil- regular, then N(I) ⊆ N(L)
Corollary
Let L be a Lie algebra over a field F. Then every minimum ideal of L is abelian, simple, or irregular.
Theorem 2
Let L be a Lie algebra.
a. If L is a solvable primitive Lie algebra then all core-free maximal subalgebras are conjugate.
b. If A is a self-centralizing minimal ideal of a solvable Lie algebra, L, then L
is primitive, A is complemented in L, and all components are conjugate.
Theorem 3
Let L be a Lie algebra.
L is semisimple if and only if its Killing form in nondegenerate. [4], pg 107
5.1 Lie’s Three Theorems
Lie I
If G is a connected and solvable linear algebraic group defined over an algebraically closed field and p : G → GL(V ) a representation on a nonzero finite-dimensional vector space, V, then there is a one-dimensional linear subspace L of V such that p(G)(L) = L.
Lie II
Let G and H be Lie groups with Lie algebras g = Lie(G) and h = Lie(H) such that G is simply connected. If f : g → h is a morphism of Lie algebras, then there is an unique morphism F : G → H of Lie groups lifting f such that f = Lie(F ).
Lie III
The functor Lie is essentially surjective on objects. For every dimensional real Lie algebra g there is a real Lie group G such that g ∼= Lie(G). Moreover, there exists a G which is simply connected.
Lie III is arguably the most challenging of the three because it requires so much structure theory of Lie algebras to prove and does not apply to general Banach-modeled Lie groups and Lie algebras.
Lie I and Lie II relate to the infinitesimal transformations of a transformation group acting on a smooth manifold. Lie III in contrast, states the Jacobi Iden- tity for the infinitesimal transformations of a local Lie group. [5]
6 Conclusion
Lie algebras are a very complex subject that relate to many fields of study. At first glance, this subject field did not seem that intriguing or deep but with more research and understanding of its applications, Lie algebras end up being quite interesting.
Theorem 3
Let L be a Lie algebra.
L is semisimple if and only if its Killing form in nondegenerate. [4], pg 107
5.1 Lie’s Three Theorems
Lie I
If G is a connected and solvable linear algebraic group defined over an algebraically closed field and p : G → GL(V ) a representation on a nonzero finite-dimensional vector space, V, then there is a one-dimensional linear subspace L of V such that p(G)(L) = L.
Lie II
Let G and H be Lie groups with Lie algebras g = Lie(G) and h = Lie(H) such that G is simply connected. If f : g → h is a morphism of Lie algebras, then there is an unique morphism F : G → H of Lie groups lifting f such that f = Lie(F ).
Lie III
The functor Lie is essentially surjective on objects. For every dimensional real Lie algebra g there is a real Lie group G such that g ∼= Lie(G). Moreover, there exists a G which is simply connected.
Lie III is arguably the most challenging of the three because it requires so much structure theory of Lie algebras to prove and does not apply to general Banach-modeled Lie groups and Lie algebras.
Lie I and Lie II relate to the infinitesimal transformations of a transformation group acting on a smooth manifold. Lie III in contrast, states the Jacobi Iden- tity for the infinitesimal transformations of a local Lie group. [5]
6 Conclusion
Lie algebras are a very complex subject that relate to many fields of study. At first glance, this subject field did not seem that intriguing or deep but with more research and understanding of its applications, Lie algebras end up being quite interesting.
Due to its wide use, it was difficult to give a brief summary on the topics it
related to because so many of the theorems and proofs that showcase Lie algebras would need a lot of explanation due to their varied and advanced material.
For example, the Weyl group is heavily influenced by Lie algebras yet they were
not discussed in this paper due to their complexity. Another topic of note that
was not brought up are Free Lie algebras which are quite interesting to learn
about.
In short, a person who studies higher level math or physics would be hard pressed to not run into Lie algebras at one point in time or another– even if they did not realize it.
As a side note, writing math papers is a lot more challenging than anticipated and it will be a long while before I complain about one of my textbooks being difficult to read. Also, it is very satisfying to spend hours writing strange code and then for it to turn into beautiful math. The least fun aspect of this project would have to have been reading hundreds of pages on a topic and learning so much about it and feeling like you understand it pretty well but do not really know what to write because none of your peers spent days reading about how Lie algebras affect quantum mechanics. The end.
In short, a person who studies higher level math or physics would be hard pressed to not run into Lie algebras at one point in time or another– even if they did not realize it.
As a side note, writing math papers is a lot more challenging than anticipated and it will be a long while before I complain about one of my textbooks being difficult to read. Also, it is very satisfying to spend hours writing strange code and then for it to turn into beautiful math. The least fun aspect of this project would have to have been reading hundreds of pages on a topic and learning so much about it and feeling like you understand it pretty well but do not really know what to write because none of your peers spent days reading about how Lie algebras affect quantum mechanics. The end.
Bibliography
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[1] H. Samelson. Notes on Lie Algebras. (Stanford, California, 1989).
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[2] D. Towers. Chief Factors of Lie Algebras. (Lancaster University, England, 2016).
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[3] H. Nishimura. The Jacobi Identity beyond Lie Algebras. (University of
Tsukuba, Japan, 2009).
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[4] J. Bernstein. Lectures on Lie Algebras. (Stockholm University, Sweden
2012).
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[5] S. Sternberg. Lie Algebras. (Harvard, Massachusetts, 2004).
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[6] A. Kirillov, Jr. Introduction to Lie Groups and Lie Algebras. (Stony Brook University, New York, 2008).
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[7] H. Georgi. Lie Algebras in Particle Physics. (Ebook, Westview Press, 1999).
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[8] A. Alexanderian Matrix Lie groups and their Lie algebras (University of
Texas, Texas, 2013).
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[9] T. Hungerford. Abstract Algebra: An Introduction (Saint Louis University,
Missouri, 2014).
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