Hellenica World

# .

V. de Paiva. "A dialectica-like model of linear logic". In Proc. Conf. on Category Theory and Computer Science, Springer-Verlag Lecture Notes in Computer Science 389, pp. 341–356, Manchester, September 1989.

Let W be the Weyl group of a semisimple Lie algebra $$\mathfrak{g}$$ (associate to fixed choice of a Cartan subalgebra $$\mathfrak{h})$$. Assume that a set of simple roots in $$\mathfrak{h}^*$$ is chosen.

The affine action (also called the dot action) of the Weyl group on the space $$\mathfrak{h}^*$$ is

$$w\cdot \lambda:=w(\lambda+\delta)-\delta$$

where $$\delta$$is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.

References

Baston, Robert J.; Eastwood, Michael G. (1989), The Penrose Transform: its Interaction with Representation Theory, Oxford University Press.

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