sl(6), type \(A^{1}_5\)
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.
Semisimple subalgebras.
Page generated by the calculator project.
Lie algebra type: A^{1}_5.
Weyl group size: 720.
A drawing of the root system in its corresponding Coxeter plane. Computations were carried out as explained by John Stembridge.
The darker red dots can be dragged with the mouse to rotate the picture.
The grey lines are the edges of the Weyl chamber.
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The root system has 30 elements.
Simple basis coordinatesEpsilon coordinatesReflection w.r.t. root
(-1, -1, -1, -1, -1)-e_{1}+e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
(0, -1, -1, -1, -1)-e_{2}+e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(-1, -1, -1, -1, 0)-e_{1}+e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 0, -1, -1, -1)-e_{3}+e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(0, -1, -1, -1, 0)-e_{2}+e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(-1, -1, -1, 0, 0)-e_{1}+e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 0, 0, -1, -1)-e_{4}+e_{6}\(s_{4}s_{5}s_{4}\)
(0, 0, -1, -1, 0)-e_{3}+e_{5}\(s_{3}s_{4}s_{3}\)
(0, -1, -1, 0, 0)-e_{2}+e_{4}\(s_{2}s_{3}s_{2}\)
(-1, -1, 0, 0, 0)-e_{1}+e_{3}\(s_{1}s_{2}s_{1}\)
(0, 0, 0, 0, -1)-e_{5}+e_{6}\(s_{5}\)
(0, 0, 0, -1, 0)-e_{4}+e_{5}\(s_{4}\)
(0, 0, -1, 0, 0)-e_{3}+e_{4}\(s_{3}\)
(0, -1, 0, 0, 0)-e_{2}+e_{3}\(s_{2}\)
(-1, 0, 0, 0, 0)-e_{1}+e_{2}\(s_{1}\)
(1, 0, 0, 0, 0)e_{1}-e_{2}\(s_{1}\)
(0, 1, 0, 0, 0)e_{2}-e_{3}\(s_{2}\)
(0, 0, 1, 0, 0)e_{3}-e_{4}\(s_{3}\)
(0, 0, 0, 1, 0)e_{4}-e_{5}\(s_{4}\)
(0, 0, 0, 0, 1)e_{5}-e_{6}\(s_{5}\)
(1, 1, 0, 0, 0)e_{1}-e_{3}\(s_{1}s_{2}s_{1}\)
(0, 1, 1, 0, 0)e_{2}-e_{4}\(s_{2}s_{3}s_{2}\)
(0, 0, 1, 1, 0)e_{3}-e_{5}\(s_{3}s_{4}s_{3}\)
(0, 0, 0, 1, 1)e_{4}-e_{6}\(s_{4}s_{5}s_{4}\)
(1, 1, 1, 0, 0)e_{1}-e_{4}\(s_{1}s_{2}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 0)e_{2}-e_{5}\(s_{2}s_{3}s_{4}s_{3}s_{2}\)
(0, 0, 1, 1, 1)e_{3}-e_{6}\(s_{3}s_{4}s_{5}s_{4}s_{3}\)
(1, 1, 1, 1, 0)e_{1}-e_{5}\(s_{1}s_{2}s_{3}s_{4}s_{3}s_{2}s_{1}\)
(0, 1, 1, 1, 1)e_{2}-e_{6}\(s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}\)
(1, 1, 1, 1, 1)e_{1}-e_{6}\(s_{1}s_{2}s_{3}s_{4}s_{5}s_{4}s_{3}s_{2}s_{1}\)
Comma delimited list of roots: (-1, -1, -1, -1, -1), (0, -1, -1, -1, -1), (-1, -1, -1, -1, 0), (0, 0, -1, -1, -1), (0, -1, -1, -1, 0), (-1, -1, -1, 0, 0), (0, 0, 0, -1, -1), (0, 0, -1, -1, 0), (0, -1, -1, 0, 0), (-1, -1, 0, 0, 0), (0, 0, 0, 0, -1), (0, 0, 0, -1, 0), (0, 0, -1, 0, 0), (0, -1, 0, 0, 0), (-1, 0, 0, 0, 0), (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1), (1, 1, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 1, 1, 0), (0, 0, 0, 1, 1), (1, 1, 1, 0, 0), (0, 1, 1, 1, 0), (0, 0, 1, 1, 1), (1, 1, 1, 1, 0), (0, 1, 1, 1, 1), (1, 1, 1, 1, 1) The resulting Lie bracket pairing table follows.
Type A^{1}_5.The letter \(\displaystyle h\) stands for elements of the Cartan subalgebra,
the letter \(\displaystyle g\) stands for the Chevalley (root space) generators of non-zero weight.
The generator \(\displaystyle h_i\) is the element of the Cartan subalgebra dual to the
i^th simple root, that is, \(\displaystyle [h_i, g] =\langle \alpha_i , \gamma\rangle g\),
where g is a Chevalley generator, \(\displaystyle \gamma\) is its weight, and
\(\displaystyle \alpha_i\) is the i^th simple root.
The Lie bracket table is too large to be rendered in LaTeX, displaying in html format instead.
roots simple coords epsilon coordinates[,]g_{-15}g_{-14}g_{-13}g_{-12}g_{-11}g_{-10}g_{-9}g_{-8}g_{-7}g_{-6}g_{-5}g_{-4}g_{-3}g_{-2}g_{-1}h_{1}h_{2}h_{3}h_{4}h_{5}g_{1}g_{2}g_{3}g_{4}g_{5}g_{6}g_{7}g_{8}g_{9}g_{10}g_{11}g_{12}g_{13}g_{14}g_{15}
(-1, -1, -1, -1, -1)-e_{1}+e_{6}g_{-15}000000000000000g_{-15}000g_{-15}g_{-14}000-g_{-13}g_{-12}00-g_{-10}g_{-9}0-g_{-6}g_{-5}-g_{-1}-h_{5}-h_{4}-h_{3}-h_{2}-h_{1}
(0, -1, -1, -1, -1)-e_{2}+e_{6}g_{-14}00000000000000g_{-15}-g_{-14}g_{-14}00g_{-14}0g_{-12}00-g_{-11}0g_{-9}0-g_{-7}0g_{-5}-g_{-2}0-h_{5}-h_{4}-h_{3}-h_{2}-g_{1}
(-1, -1, -1, -1, 0)-e_{1}+e_{5}g_{-13}0000000000-g_{-15}0000g_{-13}00g_{-13}-g_{-13}g_{-11}00-g_{-10}0g_{-8}0-g_{-6}0g_{-4}-g_{-1}0-h_{4}-h_{3}-h_{2}-h_{1}0g_{5}
(0, 0, -1, -1, -1)-e_{3}+e_{6}g_{-12}000000000g_{-15}000g_{-14}00-g_{-12}g_{-12}0g_{-12}00g_{-9}0-g_{-8}00g_{-5}-g_{-3}00-h_{5}-h_{4}-h_{3}0-g_{2}-g_{6}
(0, -1, -1, -1, 0)-e_{2}+e_{5}g_{-11}0000000000-g_{-14}000g_{-13}-g_{-11}g_{-11}0g_{-11}-g_{-11}0g_{-8}0-g_{-7}00g_{-4}-g_{-2}00-h_{4}-h_{3}-h_{2}0-g_{1}g_{5}0
(-1, -1, -1, 0, 0)-e_{1}+e_{4}g_{-10}000000-g_{-15}0000-g_{-13}000g_{-10}0g_{-10}-g_{-10}0g_{-7}0-g_{-6}00g_{-3}-g_{-1}00-h_{3}-h_{2}-h_{1}00g_{4}0g_{9}
(0, 0, 0, -1, -1)-e_{4}+e_{6}g_{-9}00000g_{-15}00g_{-14}000g_{-12}0000-g_{-9}g_{-9}g_{-9}000g_{-5}-g_{-4}000-h_{5}-h_{4}00-g_{3}0-g_{7}-g_{10}
(0, 0, -1, -1, 0)-e_{3}+e_{5}g_{-8}000000000g_{-13}-g_{-12}00g_{-11}00-g_{-8}g_{-8}g_{-8}-g_{-8}00g_{-4}-g_{-3}000-h_{4}-h_{3}00-g_{2}g_{5}-g_{6}00
(0, -1, -1, 0, 0)-e_{2}+e_{4}g_{-7}000000-g_{-14}0000-g_{-11}00g_{-10}-g_{-7}g_{-7}g_{-7}-g_{-7}00g_{-3}-g_{-2}000-h_{3}-h_{2}00-g_{1}g_{4}00g_{9}0
(-1, -1, 0, 0, 0)-e_{1}+e_{3}g_{-6}000-g_{-15}000-g_{-13}0000-g_{-10}00g_{-6}g_{-6}-g_{-6}00g_{-2}-g_{-1}000-h_{2}-h_{1}000g_{3}00g_{8}0g_{12}
(0, 0, 0, 0, -1)-e_{5}+e_{6}g_{-5}00g_{-15}0g_{-14}00g_{-12}000g_{-9}000000-g_{-5}2g_{-5}0000-h_{5}000-g_{4}00-g_{8}0-g_{11}-g_{13}
(0, 0, 0, -1, 0)-e_{4}+e_{5}g_{-4}00000g_{-13}00g_{-11}0-g_{-9}0g_{-8}0000-g_{-4}2g_{-4}-g_{-4}000-h_{4}000-g_{3}g_{5}0-g_{7}0-g_{10}00
(0, 0, -1, 0, 0)-e_{3}+e_{4}g_{-3}000000-g_{-12}00g_{-10}0-g_{-8}0g_{-7}00-g_{-3}2g_{-3}-g_{-3}000-h_{3}000-g_{2}g_{4}0-g_{6}0g_{9}000
(0, -1, 0, 0, 0)-e_{2}+e_{3}g_{-2}000-g_{-14}000-g_{-11}0000-g_{-7}0g_{-6}-g_{-2}2g_{-2}-g_{-2}000-h_{2}000-g_{1}g_{3}000g_{8}00g_{12}0
(-1, 0, 0, 0, 0)-e_{1}+e_{2}g_{-1}0-g_{-15}00-g_{-13}000-g_{-10}0000-g_{-6}02g_{-1}-g_{-1}000-h_{1}0000g_{2}000g_{7}00g_{11}0g_{14}
(0, 0, 0, 0, 0)0h_{1}-g_{-15}g_{-14}-g_{-13}0g_{-11}-g_{-10}00g_{-7}-g_{-6}000g_{-2}-2g_{-1}000002g_{1}-g_{2}000g_{6}-g_{7}00g_{10}-g_{11}0g_{13}-g_{14}g_{15}
(0, 0, 0, 0, 0)0h_{2}0-g_{-14}0g_{-12}-g_{-11}00g_{-8}-g_{-7}-g_{-6}00g_{-3}-2g_{-2}g_{-1}00000-g_{1}2g_{2}-g_{3}00g_{6}g_{7}-g_{8}00g_{11}-g_{12}0g_{14}0
(0, 0, 0, 0, 0)0h_{3}000-g_{-12}0-g_{-10}g_{-9}-g_{-8}-g_{-7}g_{-6}0g_{-4}-2g_{-3}g_{-2}0000000-g_{2}2g_{3}-g_{4}0-g_{6}g_{7}g_{8}-g_{9}g_{10}0g_{12}000
(0, 0, 0, 0, 0)0h_{4}00-g_{-13}0-g_{-11}g_{-10}-g_{-9}-g_{-8}g_{-7}0g_{-5}-2g_{-4}g_{-3}000000000-g_{3}2g_{4}-g_{5}0-g_{7}g_{8}g_{9}-g_{10}g_{11}0g_{13}00
(0, 0, 0, 0, 0)0h_{5}-g_{-15}-g_{-14}g_{-13}-g_{-12}g_{-11}0-g_{-9}g_{-8}00-2g_{-5}g_{-4}00000000000-g_{4}2g_{5}00-g_{8}g_{9}0-g_{11}g_{12}-g_{13}g_{14}g_{15}
(1, 0, 0, 0, 0)e_{1}-e_{2}g_{1}-g_{-14}0-g_{-11}00-g_{-7}000-g_{-2}0000h_{1}-2g_{1}g_{1}0000g_{6}0000g_{10}000g_{13}00g_{15}0
(0, 1, 0, 0, 0)e_{2}-e_{3}g_{2}0-g_{-12}00-g_{-8}000-g_{-3}g_{-1}000h_{2}0g_{2}-2g_{2}g_{2}00-g_{6}0g_{7}0000g_{11}000g_{14}000
(0, 0, 1, 0, 0)e_{3}-e_{4}g_{3}000-g_{-9}0g_{-6}0-g_{-4}g_{-2}000h_{3}000g_{3}-2g_{3}g_{3}00-g_{7}0g_{8}0-g_{10}00g_{12}000000
(0, 0, 0, 1, 0)e_{4}-e_{5}g_{4}00g_{-10}0g_{-7}0-g_{-5}g_{-3}000h_{4}00000g_{4}-2g_{4}g_{4}00-g_{8}0g_{9}0-g_{11}00-g_{13}00000
(0, 0, 0, 0, 1)e_{5}-e_{6}g_{5}g_{-13}g_{-11}0g_{-8}00g_{-4}000h_{5}0000000g_{5}-2g_{5}000-g_{9}000-g_{12}00-g_{14}0-g_{15}00
(1, 1, 0, 0, 0)e_{1}-e_{3}g_{6}-g_{-12}0-g_{-8}00-g_{-3}000h_{2}+h_{1}000g_{1}-g_{2}-g_{6}-g_{6}g_{6}0000g_{10}0000g_{13}000g_{15}000
(0, 1, 1, 0, 0)e_{2}-e_{4}g_{7}0-g_{-9}00-g_{-4}g_{-1}00h_{3}+h_{2}000g_{2}-g_{3}0g_{7}-g_{7}-g_{7}g_{7}0-g_{10}00g_{11}0000g_{14}000000
(0, 0, 1, 1, 0)e_{3}-e_{5}g_{8}00g_{-6}-g_{-5}g_{-2}00h_{4}+h_{3}000g_{3}-g_{4}000g_{8}-g_{8}-g_{8}g_{8}0-g_{11}00g_{12}-g_{13}000000000
(0, 0, 0, 1, 1)e_{4}-e_{6}g_{9}g_{-10}g_{-7}0g_{-3}00h_{5}+h_{4}000g_{4}-g_{5}00000g_{9}-g_{9}-g_{9}00-g_{12}000-g_{14}00-g_{15}00000
(1, 1, 1, 0, 0)e_{1}-e_{4}g_{10}-g_{-9}0-g_{-4}00h_{3}+h_{2}+h_{1}00g_{1}-g_{3}00g_{6}0-g_{7}-g_{10}0-g_{10}g_{10}0000g_{13}0000g_{15}000000
(0, 1, 1, 1, 0)e_{2}-e_{5}g_{11}0-g_{-5}g_{-1}0h_{4}+h_{3}+h_{2}00g_{2}-g_{4}00g_{7}0-g_{8}0g_{11}-g_{11}0-g_{11}g_{11}-g_{13}000g_{14}0000000000
(0, 0, 1, 1, 1)e_{3}-e_{6}g_{12}g_{-6}g_{-2}0h_{5}+h_{4}+h_{3}00g_{3}-g_{5}00g_{8}0-g_{9}000g_{12}-g_{12}0-g_{12}0-g_{14}000-g_{15}000000000
(1, 1, 1, 1, 0)e_{1}-e_{5}g_{13}-g_{-5}0h_{4}+h_{3}+h_{2}+h_{1}0g_{1}-g_{4}0g_{6}0-g_{8}0g_{10}00-g_{11}-g_{13}00-g_{13}g_{13}0000g_{15}0000000000
(0, 1, 1, 1, 1)e_{2}-e_{6}g_{14}g_{-1}h_{5}+h_{4}+h_{3}+h_{2}0g_{2}-g_{5}0g_{7}0-g_{9}0g_{11}00-g_{12}0g_{14}-g_{14}00-g_{14}-g_{15}00000000000000
(1, 1, 1, 1, 1)e_{1}-e_{6}g_{15}h_{5}+h_{4}+h_{3}+h_{2}+h_{1}g_{1}-g_{5}g_{6}0-g_{9}g_{10}00-g_{12}g_{13}000-g_{14}-g_{15}000-g_{15}000000000000000
We define the symmetric Cartan matrix
by requesting that the entry in the i-th row and j-th column
be the scalar product of the i^th and j^th roots. The symmetric Cartan matrix is:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0\\ 0 & -1 & 2 & -1 & 0\\ 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
Let the (i, j)^{th} entry of the symmetric Cartan matrix be a_{ij}.
Then we define the co-symmetric Cartan matrix as the matrix whose (i, j)^{th} entry equals 4*a_{ij}/(a_{ii}*a_{jj}). In other words, the co-symmetric Cartan matrix is the symmetric Cartan matrix of the dual root system. The co-symmetric Cartan matrix equals:
\(\displaystyle \begin{pmatrix}2 & -1 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0\\ 0 & -1 & 2 & -1 & 0\\ 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & -1 & 2\\ \end{pmatrix}\)
The determinant of the symmetric Cartan matrix is: 6
Half sum of positive roots: (5/2, 4, 9/2, 4, 5/2)= \(\displaystyle 5/2\varepsilon_{1}+3/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-3/2\varepsilon_{5}-5/2\varepsilon_{6}\)
The fundamental weights (the j^th fundamental weight has scalar product 1
with the j^th simple root times 2 divided by the root length squared,
and 0 with the remaining simple roots):
(5/6, 2/3, 1/2, 1/3, 1/6) = \(\displaystyle 5/6\varepsilon_{1}-1/6\varepsilon_{2}-1/6\varepsilon_{3}-1/6\varepsilon_{4}-1/6\varepsilon_{5}-1/6\varepsilon_{6}\)
(2/3, 4/3, 1, 2/3, 1/3) = \(\displaystyle 2/3\varepsilon_{1}+2/3\varepsilon_{2}-1/3\varepsilon_{3}-1/3\varepsilon_{4}-1/3\varepsilon_{5}-1/3\varepsilon_{6}\)
(1/2, 1, 3/2, 1, 1/2) = \(\displaystyle 1/2\varepsilon_{1}+1/2\varepsilon_{2}+1/2\varepsilon_{3}-1/2\varepsilon_{4}-1/2\varepsilon_{5}-1/2\varepsilon_{6}\)
(1/3, 2/3, 1, 4/3, 2/3) = \(\displaystyle 1/3\varepsilon_{1}+1/3\varepsilon_{2}+1/3\varepsilon_{3}+1/3\varepsilon_{4}-2/3\varepsilon_{5}-2/3\varepsilon_{6}\)
(1/6, 1/3, 1/2, 2/3, 5/6) = \(\displaystyle 1/6\varepsilon_{1}+1/6\varepsilon_{2}+1/6\varepsilon_{3}+1/6\varepsilon_{4}+1/6\varepsilon_{5}-5/6\varepsilon_{6}\)
Below is the simple basis realized in epsilon coordinates. Please note that the epsilon coordinate realizations do not have long roots of length of 2 in types G and C. This means that gramm matrix (w.r.t. the standard scalar product) of the epsilon coordinate realizations in types G and C does not equal the corresponding symmetric Cartan matrix.
(1, 0, 0, 0, 0) = \(\displaystyle \varepsilon_{1}-\varepsilon_{2}\)
(0, 1, 0, 0, 0) = \(\displaystyle \varepsilon_{2}-\varepsilon_{3}\)
(0, 0, 1, 0, 0) = \(\displaystyle \varepsilon_{3}-\varepsilon_{4}\)
(0, 0, 0, 1, 0) = \(\displaystyle \varepsilon_{4}-\varepsilon_{5}\)
(0, 0, 0, 0, 1) = \(\displaystyle \varepsilon_{5}-\varepsilon_{6}\)