30 2. Basic notions of representation theory
spread of his mental illness, possibly fueled by his opponents, who
tried to invalidate his accusations.
In the meantime, trying to assert its cultural (and eventually
political) independence from Sweden, Norway took steps to bring back
its leading intellectuals. The Norwegian National Assembly voted to
establish a personal chair in transformation group theory for Lie,
matching his high Leipzig salary. Lie was anxious to return to his
homeland, but his wife and three children did not share his nostalgia.
He eventually returned to Norway in 1898 with only a few months to
Lie “thought and wrote in grandiose terms, in a style that has
now gone out of fashion, and that would be censored by our scientific
journals”, wrote one commentator [26, p. iii]. Lie was always more
concerned with originality than with rigor. “Let us reason with con-
cepts!” he often exclaimed during his lectures and drew geometrical
pictures instead of providing analytical proofs [22, p. 244]. “With-
out Phantasy one would never become a Mathematician”, he wrote.
“[W]hat gave me a Place among the Mathematicians of our Day,
despite my Lack of Knowledge and Form, was the Audacity of my
Thinking” [56, p. 409]. Hardly lacking relevant knowledge, Lie indeed
had trouble putting his ideas into publishable form. Due to Engel’s
diligence, Lie’s research on transformation groups was summed up
in three grand volumes, but Lie never liked this ghost-written work
and preferred citing his own earlier papers [47, p. 310]. He had even
less luck with the choice of assistant to write up results on contact
transformations and partial differential equations. Felix Hausdorff’s
interests led him elsewhere, and Lie’s thoughts on these subjects were
never completely spelled out [16, p. 324]. Thus we may never discover
the “true Lie”.
2.11. Tensor products
In this subsection we recall the notion of tensor product of vector
spaces, which will be extensively used below.
Definition 2.11.1. The tensor product V ⊗ W of vector spaces V
and W over a field k is the quotient of the space V ∗ W whose basis