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Visualizing Lie Subalgebras using Root and Weight Diagrams

Author(s): 
Tevian Dray (Oregon State Univ.) and Aaron Wangberg (Winona State Univ.)

 

5. Conclusion

We have presented here methods which illustrate how root and weight diagrams can be used to visually identify the subalgebras of a given Lie algebra. While the standard methods of determining subalgebras rely upon adding, removing, or folding along nodes in a Dynkin diagram, we show here how to construct any of a Lie algebra's root or weight diagrams from its Dynkin diagram, and how to use geometric transformations to visually identify subalgebras using those weight and root diagrams. In particular, we show how these methods can be applied to algebras whose root and weight diagrams have dimensions four or greater. In addition to pointing out the erroneous inclusion of \(C_4 \subset F_4\) in [15, 16], we provide visual proof that \(C_4 \subset E_6\) and list all the subalgebras of \(E_6\). While we are primarily concerned with the subalgebras of \(E_6\), these methods can be used to find subalgebras of any rank \(l\) algebra.

Endnotes

1There are two minimal representations of \(A_2\), only one of which is shown in Figure 2. The second minimal weight diagram of \(A_2\) is similar, but rotated \(180^\circ\) from the one shown. We omit this diagram, since it contributes no new information to the determination of subalgebras, which is our primary goal.

2For algebras of rank 6 and lower, the exceptions are that \(B_n = so(2n+1)\) and \(C_n=sp(2 \cdot n)\) have the same dimension for each rank \(n\), and that \(B_6\), \(C_6\), and \(E_6\) all have dimension 78.

3The distance between adjacent weights in the infinite lattice is less than or equal to the length of each root vector. Root vectors do not always connect adjacent weights, but often skip over them.

4 An equivalent projection \(\mathbb{R}^6 o \mathbb{R}^3: (x,y,z,u_1,u_2,u_3) o (x+s_1 u_1 + s_2 u_2 + s_3 u_3, y, z)\) with \(s_1 > \eta_2 s_2 > \eta_3 s_3\) for sufficiently large \(\eta_2\) and \(\eta_3\) will string a 6-dimensional diagram along one axis. The \(u_1\) coordinate will separate the different 5-dimensional slices, with \(s_1 u_1\) moving these different slices very apart. The \(u_2\) and \(u_3\) coordinates will locally separate the 4-dimensional and 3-dimensional slices along the \(x\) axis, but sufficiently small \(s_2\) and \(s_3\) will keep the subslices of one 5-dimensional slice from interfering with another 5-dimensional slice. This method generalizes to \(n\)-dimensional diagrams, but creates a very long string of the resulting diagrams.

5 The large number of grey lines in a diagram can hide the important roots within each slice in addition to causing computational overload.

6 While the root vectors in the octahedron and the \(C_3\) root diagram appear to intersect, they do not terminate or start from any common vertex.

References

[1] Nathan Jacobson. Lie Algebras, and Representations: An Elementary Introduction. 1st edition, 2006.
[2] J. F. Cornwell. Group Theory in Physics: An Introduction. ACADEMIC PRESS, San Diego, California, 1997.
[3] Wikipedia. Lie algebra - wikipedia, the free encyclopedia, 2006. Available at http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=93493269 [Online; accessed 31-December-2006].
[4] Corinne A. Manogue and Jorg Schray. Finite Lorentz Transformations, Automorphisms, and Division Algebras. J. Math. Phys., 34:3746-3767, 1993.
[5] E. Corrigan and T. J. Hollowood. A String Construction of a Commutative Non-Associative Algebra related to the Exceptional Jordan Algebra. Physics Letters B, 203:47-51, March 1988.
[6] John C. Baez. The Octonions. Bull. Amer. Math. Soc., 39:145-205, 2002. Also available as http://math.ucr.edu/home/baez/octonions/.
[7] Susumu Okubo. Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press, Cambridge, 1995. See also http://www.ams.org/mathscinet-getitem?mr=96j:81052.
[8] The Atlas Project. Representations of E8, 2007. Available at http://aimath.org/E8/.
[9] E. B. Dynkin. Semi-simple subalgebras of semi-simple lie algebras. Am. Math. Soc. Trans., No. 1:1-143, 1950.
[10] J. Patera S. Kass, R. V. Moody and R. Slansky. Affine Lie Algebras, Weight Multiplicities, and Branching Rules, volume 1. University of California Press, Los Angeles, California, 1990.
[11] Zome tools. Available at http://www.zometools.com.
[12] Ulf H. Danielsson and Bo Sundborg. Exceptional Equivalences in N=2 Supersymmetric Yang-Mills Theory. Physics Letters B, 370:83, 1996. Also available as http://arXiv.org/abs/hep-th/9511180.
[13] Aaron Wangberg. The Structure of \(E_6\). Ph. D. thesis, Oregon State University, 2007.
[14] Aaron Wangberg. Subalgebras of \(E_6\) using root and weight diagrams, 2006. Available at http://oregonstate.edu/~drayt/JOMA/subalgebra_frameset.htm.
[15] Robert Gilmore. Lie Groups, Lie Algebras, and Some of Their Applications. Wiley, 1974. Reprinted by Dover Publications, Mineola, New York, 2005.
[16] B. L. van der Waerden. Die Klassifikation der einfachen Lieschen Gruppen. Mathematische Zeitschrift, 37:446-462, 1933.

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