Corrections in infinite dimensions

Several well-known formulas involving reflection groups of finite-dimensional algebraic systems break down in infinite dimensions, but there is often a predictable way to correct them. Oxford Mathematician Thomas Oliver talks about his research getting to grips with what structures underlie the mysterious correction process.

"In algebraic language, a reflection is an order two symmetry. What this means in practical terms is that to invert a reflection, one needs to reflect again. Reflection groups are formed by composing reflections in different mirrors, the effect of which can be visualised using a kaleidoscope.

A Weyl group is a special type of reflection group. Weyl groups describe the reflective symmetries of root systems, which are configurations of vectors with prescribed geometric properties. Root systems and their Weyl groups played a pivotal role in a key achievement of modern mathematics, namely the classification of semisimple finite-dimensional Lie algebras.

There are more general theories of infinite-dimensional Lie algebras, which tentatively have applications to arithmetic and physics. The finite-dimensional theory is not exactly valid, but it can often be "corrected" in a predictable way. In fact several important equations need to be adjusted by the same correction factor.

When mathematicians see the same factors appearing in different contexts, they demand an explanation. In the case of the correction factor, their appearance is due to a new phenomenon for root systems of infinite-dimensional Lie algebras. This is the existence of imaginary roots, which have the unlikely sounding property of having negative length. This is nothing more than colourful language, in which "length" is a word for an inner product.

We calculated the correction factor as an infinite sum over the roots and analysed its "support", that is, the roots contributing non-zero terms to the sum. We found that the reflection group swapped roots around within the support, but never took one outside. That is, the support is Weyl group invariant. This corresponds to a key property of the imaginary roots: no matter what combination of reflections you try, you can never pass from an imaginary root to a real one. Because finite-dimensional Lie algebras do not have any imaginary roots, our analysis in fact gives a new proof of several classical formulas.

The research was carried out with Kyu-Hwan Lee and Dongwen Liu."

Image above: the action of reflection groups can be visualised like a kaleidoscope.