From: "j.e.mebius" Subject: Lie groups/algebras D4 and F4 Date: 5 Mar 2001 08:40:04 -0600 Newsgroups: sci.math.research Summary: Minimal representations of Lie group F4 A QUESTION ON THE LIE ALGEBRAS AND LIE GROUPS SO(8) AND F4 Johan E. Mebius (mailto:j.e.mebius@its.tudelft.nl) URL of this note: http://www.xs4all.nl/~plast/So8f4.htm March 2001 Open question, at least for me: Is the exceptional Lie group F4 a subgroup of some group-theoretic product of two copies of SO(8) or Spin(8)? Related question: Can the coset space of a subgroup SO(2)^4 of SO(8) x SO(8) (or of Spin(8) x Spin(8)) be turned into F4 in some way? The root system of the exceptional Lie algebra f4 consists of 48 root vectors of two different sizes, 24 long ones and 24 short ones. Let (e1, e2, e3, e4) be an orthonormal basis of R4. Then the the set Rl consisting of the 24 vectors � ei � ej (i, j = 1, 2, 3, 4) and the set Rs consisting of the 8 vectors � ei together with the 16 vectors (� e1 � e2 � e3 � e4) / 2 make up the root vectors of f4. The set Rs can be transformed into Rl by a 4D rotation combined with a sqrt2 : 1 dilatation with the origin as a centre. Both sets Rs and Rl in their own right are a root system of the Lie algebra d4 = so(8). (See F4roots.htm). This observation is believed to be new to the literature on root systems. It leads to the questions at the top of this article. Look at ranks and numbers of dimensions: SO(8) is 28D and has rank 4; SO(8) x SO(8) is 56D and has rank 8. Take a subgroup SO(2)^4 of SO(8) x SO(8). It is a 4D commutative group, so its rank equals 4. Inside SO(8) x SO(8) it yields a 52D coset space which presumably would have rank 4 whenever it could be made into a group. This note was inspired by Chapter 9 of [1], where a subdivision of the full root system of f4 into the set Ra consisting of � ei � ej and � ei (i, j = 1, 2, 3, 4) and the set Rb consisting of (� e1 � e2 � e3 �e4) / 2 is presented, corresponding to the well-known inclusion of Spin(9) in F4 with the octonion projective plane as a coset space. The set Ra is the root system of so(9), which is a maximal Lie subalgebra of f4; the set Rb corresponds to the coset space of so(9) in f4. The set Rb is not the root system of any simple Lie algebra. The corresponding group coset space cannot be made into a simple Lie group. Remark: The root system of G2 is the union of two copies of the root system of SU(3). In this respect it resembles the root system of F4. So one may ask analogous questions about G2, SU(3) x SU(3) and the coset space of a subgroup SO(2) x SO(2) of SU(3) x SU(3). Literature [1] J.F.Adams: Lectures on exceptional Lie groups. Eds.: Zafer Mahmud and Mamoru Mimura. The University of Chicago Press, 1996, ISBN 0-226-00526-7 [reformatted --djr] ============================================================================== From: Linus Kramer Subject: Re: Lie groups/algebras D4 and F4 Date: 5 Mar 2001 10:20:03 -0600 Newsgroups: sci.math.research No, this is not possible. Let F_4 --> Spin(8) x Spin(8) be any Lie group homomorphism. Combine it with the projection onto the first or second factor, so you get a representation of F_4 on R^8. The only such representation of F_4 is the trivial one, so each of the two arrows F_4 --> Spin(8) x Spin(8) --> SO(8) is trivial. The smallest nontrivial real F_4-module has dimension 26. Regards, Linus Kramer -- Linus Kramer Mathematisches Institut Universitaet Wuerzburg Am Hubland 97074 Wuerzburg Germany E-mail: kramer@mathematik.uni-wuerzburg.de http://www.mathematik.uni-wuerzburg.de/~kramer