[math-fun] Lie group terminology
I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra. When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me. I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined. So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was. The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones. This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was. So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold". Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum. Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'." Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me. Help.
For a simpler example, replace 'E8' with 'SL(2)'. This can be either a finite group (when you're over a finite field such as F_5) or an infinite group with manifold structure (when you're over the reals or complexes). Exactly the same applies to E8: over finite fields, you get big (simple!) finite groups; over the reals, you get some infinite group with manifold structure. And over the (((field of one element))), you get the Weyl group of E8, which is the group of symmetries of the E8 root system, or E8 polytope, or the origin-fixing symmetries of the E8 lattice.
Sent: Saturday, October 06, 2018 at 3:21 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Lie group terminology
I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra.
When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me.
I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined.
So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was.
The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones.
This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was.
So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold".
Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum.
Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'."
Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me.
Help. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Ouch -- I didn't realise the etymology of the triple parentheses, and then just Googled it only to find it has political connotations. On the occasions I'd seen it used, I thought it indicated that the enclosed text was meant to be said very quietly (i.e. the opposite of using *emphasis*). Knowing its origin, I redact the use of that punctuation in the e-mail below. Apologies for my ignorance. -- APG.
Sent: Saturday, October 06, 2018 at 3:59 PM From: "Adam P. Goucher" <apgoucher@gmx.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Lie group terminology
For a simpler example, replace 'E8' with 'SL(2)'. This can be either a finite group (when you're over a finite field such as F_5) or an infinite group with manifold structure (when you're over the reals or complexes).
Exactly the same applies to E8: over finite fields, you get big (simple!) finite groups; over the reals, you get some infinite group with manifold structure.
And over the (((field of one element))), you get the Weyl group of E8, which is the group of symmetries of the E8 root system, or E8 polytope, or the origin-fixing symmetries of the E8 lattice.
Sent: Saturday, October 06, 2018 at 3:21 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Lie group terminology
I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra.
When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me.
I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined.
So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was.
The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones.
This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was.
So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold".
Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum.
Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'."
Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me.
Help. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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To get you used to the geometric thinking surrounding Lie groups I'd recommend the lovely monograph "Finite Reflection Groups" by Clark Benson and Larry Grove: https://www.springer.com/us/book/9780387960821 Also, the following free lecture notes are pretty good: http://homepages.warwick.ac.uk/~masdf/modular/Reflection_Groups.pdf Victor On Sat, Oct 6, 2018 at 11:10 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
Ouch -- I didn't realise the etymology of the triple parentheses, and then just Googled it only to find it has political connotations. On the occasions I'd seen it used, I thought it indicated that the enclosed text was meant to be said very quietly (i.e. the opposite of using *emphasis*).
Knowing its origin, I redact the use of that punctuation in the e-mail below. Apologies for my ignorance.
-- APG.
Sent: Saturday, October 06, 2018 at 3:59 PM From: "Adam P. Goucher" <apgoucher@gmx.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Lie group terminology
For a simpler example, replace 'E8' with 'SL(2)'. This can be either a finite group (when you're over a finite field such as F_5) or an infinite group with manifold structure (when you're over the reals or complexes).
Exactly the same applies to E8: over finite fields, you get big (simple!) finite groups; over the reals, you get some infinite group with manifold structure.
And over the (((field of one element))), you get the Weyl group of E8, which is the group of symmetries of the E8 root system, or E8 polytope, or the origin-fixing symmetries of the E8 lattice.
Sent: Saturday, October 06, 2018 at 3:21 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Lie group terminology
I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra.
When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me.
I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined.
So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was.
The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones.
This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was.
So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold".
Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum.
Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'."
Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me.
Help. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Also Terry Tao's course notes on Hilbert's 5th problem: https://terrytao.files.wordpress.com/2012/03/hilbert-book.pdf On Sat, Oct 6, 2018 at 12:00 PM Victor Miller <victorsmiller@gmail.com> wrote:
To get you used to the geometric thinking surrounding Lie groups I'd recommend the lovely monograph "Finite Reflection Groups" by Clark Benson and Larry Grove: https://www.springer.com/us/book/9780387960821
Also, the following free lecture notes are pretty good: http://homepages.warwick.ac.uk/~masdf/modular/Reflection_Groups.pdf
Victor
On Sat, Oct 6, 2018 at 11:10 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
Ouch -- I didn't realise the etymology of the triple parentheses, and then just Googled it only to find it has political connotations. On the occasions I'd seen it used, I thought it indicated that the enclosed text was meant to be said very quietly (i.e. the opposite of using *emphasis*).
Knowing its origin, I redact the use of that punctuation in the e-mail below. Apologies for my ignorance.
-- APG.
Sent: Saturday, October 06, 2018 at 3:59 PM From: "Adam P. Goucher" <apgoucher@gmx.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Lie group terminology
For a simpler example, replace 'E8' with 'SL(2)'. This can be either a finite group (when you're over a finite field such as F_5) or an infinite group with manifold structure (when you're over the reals or complexes).
Exactly the same applies to E8: over finite fields, you get big (simple!) finite groups; over the reals, you get some infinite group with manifold structure.
And over the (((field of one element))), you get the Weyl group of E8, which is the group of symmetries of the E8 root system, or E8 polytope, or the origin-fixing symmetries of the E8 lattice.
Sent: Saturday, October 06, 2018 at 3:21 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Lie group terminology
I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra.
When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me.
I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined.
So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was.
The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones.
This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was.
So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold".
Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum.
Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'."
Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me.
Help. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Tut, tut --- uncle don't allow pol-i-ticking here! It is noteworthy the extent to which apparently essentially continuous notions can meaningfully be reinterpreted over discrete or finite domains. Algebraic geometry is one instance: eg. elliptic curves over finite fields, as applied to cryptography. I wonder whether useful insights result from regarding E8 etc. over |F_q as curves ... WFL On 10/6/18, Adam P. Goucher <apgoucher@gmx.com> wrote:
Ouch -- I didn't realise the etymology of the triple parentheses, and then just Googled it only to find it has political connotations. On the occasions I'd seen it used, I thought it indicated that the enclosed text was meant to be said very quietly (i.e. the opposite of using *emphasis*).
Knowing its origin, I redact the use of that punctuation in the e-mail below. Apologies for my ignorance.
-- APG.
Sent: Saturday, October 06, 2018 at 3:59 PM From: "Adam P. Goucher" <apgoucher@gmx.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Lie group terminology
For a simpler example, replace 'E8' with 'SL(2)'. This can be either a finite group (when you're over a finite field such as F_5) or an infinite group with manifold structure (when you're over the reals or complexes).
Exactly the same applies to E8: over finite fields, you get big (simple!) finite groups; over the reals, you get some infinite group with manifold structure.
And over the (((field of one element))), you get the Weyl group of E8, which is the group of symmetries of the E8 root system, or E8 polytope, or the origin-fixing symmetries of the E8 lattice.
Sent: Saturday, October 06, 2018 at 3:21 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Lie group terminology
I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra.
When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me.
I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined.
So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was.
The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones.
This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was.
So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold".
Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum.
Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'."
Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me.
Help. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I don't completely understand all of this (though I would like to---want a study partner?), but I can help with one piece of the puzzle. The prototypical examples of Lie groups are matrix groups, subgroups of GL(n), the group of invertible n x n matrices. I think it's almost, but not quite, true, that all Lie groups are subgroups of matrix groups. (the "almost" is that I think there are a few (families of?) Lie groups that are double-covers of matrix groups, but aren't matrix groups themselves. Some of the well-known matrix groups, such as SO(n) (n x n orthogonal matrices with determinant 1) are simple (that is, simple as Lie groups, rather than as abstract groups---I don't know the definition of a simple Lie group, but it's analogous but not identical to the definition of a simple abstract group. Now here's the remarkable fact. Or at least it's remarkable to me---maybe it's obvious to someone who really understands all of this. When I say SO(n) above, I really should be saying SO(n, R), that is, orthogonal n x n matrices *of real numbers* with determinant 1. If you take the corresponding finite groups, that is SO(n, F), for a finite field F, the resulting finite groups are simple as abstract groups. There are multiple families of simple Lie groups like SO(n, R), and in all cases, by replacing R by a finite field, we get simple finite groups. I don't know the classification theorem for simple Lie groups, but it's considerably simpler, and was proved considerably earlier, than the corresponding theorem for finite groups. In addition to the infinite families of simple Lie groups, there are a few extra, exceptional simple Lie groups, like E6, E7, E8, and F4. These also have some way they can be defined that allows the replacing of R with a finite field to arrive at finite simple groups. While E8 is a single simple Lie group, it gives rise to a family of finite simple groups, because we have a choice of which finite field to use. The finite simple groups that arise in this way from simple Lie groups, whether from the infinite families of simple Lie groups like SO(n) or the finite number of exceptional simple Lie groups like E8, are called "groups of Lie type"; they are like simple Lie groups, only finite. Finite simple groups consist of the groups of Lie type, the alternating group An of odd permutations (n > 4), and 26 or 27 additional "sporadic groups", depending on how you count. Andy On Sat, Oct 6, 2018 at 10:22 AM Allan Wechsler <acwacw@gmail.com> wrote:
I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra.
When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me.
I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined.
So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was.
The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones.
This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was.
So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold".
Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum.
Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'."
Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me.
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participants (5)
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Adam P. Goucher -
Allan Wechsler -
Andy Latto -
Fred Lunnon -
Victor Miller