Differences

This shows you the differences between two versions of the page.

--- advanced_tools:group_theory:lie_algebras [2018/04/08 12:05]
jakobadmin [Researcher]
+++ advanced_tools:group_theory:lie_algebras [2023/03/29 17:40] (current)
132.76.61.54 fur -> for
@@ Line 1: / Line 1: @@
 ====== Lie Algebras ======
-<tabbox Why is it interesting?>
-One of the most important ideas of Lie theory is that we can learn a lot about something complicated: a Lie group, by looking at something much simpler: its Lie algebra. A Lie group is a complicated mathematical object (a manifold), whereas a Lie algebra is simply a vector space. A manifold is defined as a set that looks like the good old Euclidean space in the neighborhood of every point. This tangent Euclidean space above each point can be used to define coordinates at each point. The Lie algebra $\mathfrak{g}$ of a given group $G$ is defined as the tangent space above the identity element $I$ of the group.
+<tabbox Intuitive>
+The Lie algebras consist of infinitesimally small transformations and are used in physics to describe [[basic_tools:symmetry|symmetries]]. Since arbitrarily large continuous transformations can be built up from tiny ones, almost everything that is important about continuous transformations is encoded in the infinitesimal ones.
-<blockquote>
+Formulated differently, Lie algebras describe infinitesimal symmetries.
-**Experiment tells us more directly about the Lie algebra of G than about G itself.** When I say that G contains the subgroup SU(3) X SU(2) x U(1), I really mean only that the Lie algebra of G contains that of SU(3) X SU(2) X U(1); there is no claim about the global form of G. For the same reason, in later comments I will not be very precise in distinguishing different groups that have the same Lie algebra.
-<cite>from [[http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0267.0306.ocr.pdf|Geometry and Physics by E. Witten]]</cite>
-</blockquote>
-<blockquote>
-**In high-energy theory. we tend to focus on the Lie algebra of a group and ignore its global structure;** for example, we indiscriminately refer to the isospin group as SU(2) or SO(3). This is an especially bad habit in monopole theory, because the quantization condition is sensitive to the global structure of G; the allowed set of monopoles is different for SU(2) and SO(3). For example, let us suppose that Q is proportional to $I_3$ , the generator of rotations about the third axis in isospin space. If G is SO(3), Q can be any half-integral multiple of $I_3$, because a rotation by $2\pi$ is the identity. If G is SU(2), though, only integral multiples are allowed, because a rotation by $4\pi$ is needed to get back to the identity.
-<cite>The Magnetic Monopole Fifty Years Later by Sidney R. Coleman </cite>
-</blockquote>
-<blockquote>
-[P]erturbative effects depend only on the Lie algebra.
-<cite>Global structure of the standard model, anomalies, and charge quantization by Joseph Huck</cite>
-</blockquote>
-<tabbox Layman>
-<note tip>
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
-</note>
-<tabbox Student>
+<tabbox Concrete>
 Lie groups describe continuous symmetries. Therefore, we can consider group elements that are arbitrarily close to the identity element. In mathematical terms, we denote a transformation really close to the identity element (which changes nothing at all) by
@@ Line 44: / Line 24: @@
 Now the idea is that if we have a finite transformation, say a rotation by $\theta= 50^\circ$, we can describe this through the repetition of an infinitesimal (= tiny tiny tiny) transformation. Everything important about the finite transformation is encoded in the infinitesimal transformation.
-To realise this idea explicitly, we divide $\theta$ by a large number $N$ in order to make sure that we are close to the identity. This means, our transformation close to the identity transformation is now written as
+To realize this idea explicitly, we divide $\theta$ by a large number $N$ in order to make sure that we are close to the identity. This means, our transformation close to the identity transformation is now written as
 \begin{equation} g(\theta)=I+ \frac{\theta}{N} X .\end{equation}
@@ Line 70: / Line 50: @@
 ** Definition:**
-<WRAP tip> For a matrix Lie group $G$ the corresponding Lie algebra  $\mathfrak{g}$ can be defined as the set of matrices $X$ that yield $e^{tX} \in G$ für $t \in \mathbb{R}$.</WRAP>
+<WRAP tip> For a matrix Lie group $G$ the corresponding Lie algebra  $\mathfrak{g}$ can be defined as the set of matrices $X$ that yield $e^{tX} \in G$ for $t \in \mathbb{R}$.</WRAP>
 **The Natural Product of a Lie Algebra:**
@@ Line 88: / Line 68: @@
 <WRAP tip> The natural product of a Lie algebra is not ordinary matrix multiplication, but the Lie bracket.</WRAP>
-----
-  * [[http://jakobschwichtenberg.com/lie-algebra-able-describe-group/|How is a Lie Algebra able to describe a Group?]]
-  * [[http://www.wetsavannaanimals.net/wordpress/what-information-about-a-lie-group-is-not-encoded-in-its-lie-algebra/|What Information About a Lie Group is Not Encoded in its Lie Algebra?]]
@@ Line 104: / Line 79: @@
 </blockquote>
-<tabbox Researcher>
+<tabbox Abstract>
+  * [[http://jakobschwichtenberg.com/lie-algebra-able-describe-group/|How is a Lie Algebra able to describe a Group?]]
+  * [[http://www.wetsavannaanimals.net/wordpress/what-information-about-a-lie-group-is-not-encoded-in-its-lie-algebra/|What Information About a Lie Group is Not Encoded in its Lie Algebra?]]
+----
 {{ :advanced_tools:group_theory:adjoint.png?nolink |}}
-<tabbox Examples>
+<tabbox Why is it interesting?>
---> Example1#
+One of the most important ideas of Lie theory is that we can learn a lot about something complicated: a Lie group, by looking at something much simpler: its Lie algebra. A Lie group is a complicated mathematical object (a manifold), whereas a Lie algebra is simply a vector space. A manifold is defined as a set that looks like the good old Euclidean space in the neighborhood of every point. This tangent Euclidean space above each point can be used to define coordinates at each point. The Lie algebra $\mathfrak{g}$ of a given group $G$ is defined as the tangent space above the identity element $I$ of the group.
-<--
---> Example2:#
+<blockquote>
+**Experiment tells us more directly about the Lie algebra of G than about G itself.** When I say that G contains the subgroup SU(3) X SU(2) x U(1), I really mean only that the Lie algebra of G contains that of SU(3) X SU(2) X U(1); there is no claim about the global form of G. For the same reason, in later comments I will not be very precise in distinguishing different groups that have the same Lie algebra.
+<cite>from [[http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0267.0306.ocr.pdf|Geometry and Physics by E. Witten]]</cite>
-<--
+</blockquote>
-<tabbox FAQ>
+<blockquote>
-<tabbox History>
+**In high-energy theory. we tend to focus on the Lie algebra of a group and ignore its global structure;** for example, we indiscriminately refer to the isospin group as SU(2) or SO(3). This is an especially bad habit in monopole theory, because the quantization condition is sensitive to the global structure of G; the allowed set of monopoles is different for SU(2) and SO(3). For example, let us suppose that Q is proportional to $I_3$ , the generator of rotations about the third axis in isospin space. If G is SO(3), Q can be any half-integral multiple of $I_3$, because a rotation by $2\pi$ is the identity. If G is SU(2), though, only integral multiples are allowed, because a rotation by $4\pi$ is needed to get back to the identity.
+<cite>The Magnetic Monopole Fifty Years Later by Sidney R. Coleman </cite>
+</blockquote>
+<blockquote>
+[P]erturbative effects depend only on the Lie algebra.
+<cite>Global structure of the standard model, anomalies, and charge quantization by Joseph Huck</cite>
+</blockquote>
 </tabbox>

User Tools

Site Tools

Differences

Page Tools