If I am in two dimensional space, the meaning I have for the span is the usual one from linear algebra. But I do not know what it means to say the roots in a root system, R, span the inner product space, E. For two reasons:

  1. Look a the diagram of root space of B_2 (drawn in the two dimensional euclidean plane) there are 8 roots but we are in 2 dimensional space. The usual meaning of span would mean we have a linearly dependent set.

  2. I can't reconcile the idea that roots can be linear functionals acting on the cartan subalgebra as an inner product vector space but then forming a subset of a vector space that does not act on the cartan subalgebra.

I am sure I am missing something simple.

  • $\begingroup$ Roots span the vector space in which they live in the sense that they linearly generate it; that is, any vector in the space can be written- perhaps not uniquely- as a linear combination of roots. $\endgroup$ – ebrahim Dec 4 '15 at 19:01
  • $\begingroup$ @1. Span means generate, not necessarily in a linearly independent way. Especially not in this context. $\endgroup$ – Justpassingby Dec 4 '15 at 19:07
  1. Any set of vectors spans the space of all linear combinations of those vectors, whether they are linearly independent or not. If they are not, that just means that there are redundancies, so you could choose a linearly independent subset of those vectors that spans the same space.

  2. Linear functionals are themselves elements of a vector space, namely the dual space $V^*$ consisting of all linear functionals on the vector space $V$, which are linear maps from $V$ into the base field. This just means that the functionals form an abelian group under addition and can be multiplied by scalars in the base field with everything behaving correctly. The abstract definition has the roots spanning a vector space that we pull out of thin air because its identity isn't important, but we can in fact choose it to be the space of linear functionals spanned by the actual roots.


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