lpetrich
Contributor
I started my semisimple-Lie-algebra series of posts at #454
I've been asked where to start on semisimple Lie algebras. Here are two possibilities, both online and neither paywalled:
Group Theory for Unified Model Building - Richard Slansky
Semi-Simple Lie Algebras and Their Representations - Robert N. Cahn
Robert Cahn discusses possible root connections in his chapter on exceptional Lie algebras. He has a proof that a SSLA's root connections cannot contain loops. I will repeat it. For a pair of roots, one will be long (al) and one will be short (as) if they have unequal lengths. This gives us
(al.as) = - (n/2)*(as.as) = - (1/2)*(al.al)
Thus, (al.al) = n*(as.as) and (al.as)/sqrt((as.as)*(al.al)) = - (1/2)*sqrt = - (1/2)*sqrt((al.al)/(as.as))
Now consider the sum of set of normalized root vectors a/sqrt((a.a)). Its absolute square ought to be greater than zero, since the roots are linearly independent. Finding that absolute square gives
N(roots) - sum(j>i) of sqrt(nij) < N(roots) - N(connections)
For a loop, N(connections) >= N(roots), making that sum's absolute square non-positive. Thus, a SSLA contains no root-connection loops.
For some of the other constraints, we can use a sum of weighted roots: ci*ai/sqrt((ai,ai)) and then take its absolute square
Let's see how it goes for a triply-connected root.
Triple-single. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(3) - c1*c3
c1 = 2, c2 = sqrt(3), c3 = 1 give 0.
Triple-double. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(3) - c1*c3*sqrt(2)
c1 = 2, c2 = sqrt(3), c3 = sqrt(2) give -1.
Triple-triple. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(3) - c1*c3*sqrt(3)
c1 = 2, c2 = sqrt(3), c3 = sqrt(3) give -2.
Double-double. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(2) - c1*c3*sqrt(2)
c1 = sqrt(2), c2 = 1, c3 = 1 give 0.
Single-single-double. c1^2 + c2^2 + c3^2 + c4^2 - c1*c2*sqrt(2) - c1*c3 - c1*c4
c1 = 2, c2 = sqrt(2), c3 = 1, c4 = 1 give 0.
Single-single-single-single. c1^2 + c2^2 + c3^2 + c4^2 + c5^2 - c1*c2 - c1*c3 - c1*c4 - c1*c5
c1 = 2, c2 = 1, c3 = 1, c4 = 1, c5 = 1 give 0.
Consider a simple Lie algebra's roots. If two roots are triply connected, they are the only roots. If two roots are doubly connected, there are no other double connections, and the roots are connected in a straight chain. If all the roots are singly connected to other roots, then there is at most one branching point, and that is a 3-way branch.
I've been asked where to start on semisimple Lie algebras. Here are two possibilities, both online and neither paywalled:
Group Theory for Unified Model Building - Richard Slansky
Semi-Simple Lie Algebras and Their Representations - Robert N. Cahn
Robert Cahn discusses possible root connections in his chapter on exceptional Lie algebras. He has a proof that a SSLA's root connections cannot contain loops. I will repeat it. For a pair of roots, one will be long (al) and one will be short (as) if they have unequal lengths. This gives us
(al.as) = - (n/2)*(as.as) = - (1/2)*(al.al)
Thus, (al.al) = n*(as.as) and (al.as)/sqrt((as.as)*(al.al)) = - (1/2)*sqrt = - (1/2)*sqrt((al.al)/(as.as))
Now consider the sum of set of normalized root vectors a/sqrt((a.a)). Its absolute square ought to be greater than zero, since the roots are linearly independent. Finding that absolute square gives
N(roots) - sum(j>i) of sqrt(nij) < N(roots) - N(connections)
For a loop, N(connections) >= N(roots), making that sum's absolute square non-positive. Thus, a SSLA contains no root-connection loops.
For some of the other constraints, we can use a sum of weighted roots: ci*ai/sqrt((ai,ai)) and then take its absolute square
Let's see how it goes for a triply-connected root.
Triple-single. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(3) - c1*c3
c1 = 2, c2 = sqrt(3), c3 = 1 give 0.
Triple-double. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(3) - c1*c3*sqrt(2)
c1 = 2, c2 = sqrt(3), c3 = sqrt(2) give -1.
Triple-triple. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(3) - c1*c3*sqrt(3)
c1 = 2, c2 = sqrt(3), c3 = sqrt(3) give -2.
Double-double. c1^2 + c2^2 + c3^2 - c1*c2*sqrt(2) - c1*c3*sqrt(2)
c1 = sqrt(2), c2 = 1, c3 = 1 give 0.
Single-single-double. c1^2 + c2^2 + c3^2 + c4^2 - c1*c2*sqrt(2) - c1*c3 - c1*c4
c1 = 2, c2 = sqrt(2), c3 = 1, c4 = 1 give 0.
Single-single-single-single. c1^2 + c2^2 + c3^2 + c4^2 + c5^2 - c1*c2 - c1*c3 - c1*c4 - c1*c5
c1 = 2, c2 = 1, c3 = 1, c4 = 1, c5 = 1 give 0.
Consider a simple Lie algebra's roots. If two roots are triply connected, they are the only roots. If two roots are doubly connected, there are no other double connections, and the roots are connected in a straight chain. If all the roots are singly connected to other roots, then there is at most one branching point, and that is a 3-way branch.