Study Theorem

My task was to compile the name index for a book. The sentence "The study of finite dimensional $U_k$-modules $M$ is dominated by the classical result due to Lie and Study ... " offered two entries to the name index: 'Lie' and 'Study'.

The sentence spoke about the 'classical result of Study' or the 'classical theorem of Study', etc. The initials should be added, too. Simple search for phrases with the word 'Study' ended with million and more of links to various documents which could not disclose the initials of the name 'Study'.

Also, some of links led to 'Study's theorem on curves' while the sentence in the book was related with something close to the Lie groups, algebras, modules, representations, etc.

Looking for 'Study theorem' in Wikipedia the Wikipedia page Wikipedia:Missing science topics/Maths27 directed me to Google book search for 'Study's theorem'. The Google book search gave about 160 links, three of which were useful enough.

In the pages of books shown via the Google book search I read the following.

"Essays in the History of Lie Groups and Algebraic Groups" by Armand Borel

p. 11:
In 1896, G. Fano, who knew about Study's theorem through [LE] and was surely not aware of Cartan's proof, maybe not even of Cartan's thesis, gave an entirely different one in the framework of algebraic geometry, using the properties of "rational normal scrulls" [F].

p. 31:
Another push into Lie groups came, in a way, again, from the tensor calculus in [WI], but for a different reason. In 1923 E. Study, a well-known expert in invariant theory for over thirty years, published a book on invariant theory [St2].


I already could conclude that the theorem was 'E. Study theorem', and the name was 'E. Study', but it was interesting to read more.

"Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1826-1926" by Thomas Hawkins
[Chapter 7 "Lie's School and Linear Representations"]

Three Privatdozenten at Leipzig, Friedrich Engel, Friedrich Schur, and Eduard Study, turned their research interests to aspects of Lie's theory.
<...>
Then by Study's complete reducibility theorem for $\mathbf{sl}(2,\mathbb{C})$ all the possibilities for $\mathfrac{m}$ could be specified. That is, Study's theorem implies
\begin{equation}
\mathbb{C}_{n+1}=\mathcal{V}_0\oplus \mathcal{V}_1\oplus\cdots\oplus \mathcal{V}_k\oplus,
\end{equation}
where $\mathfrac{m}$ leaves each vector subspace $\mathcal{V}_i$ invariant but leaves no nontrivial subspace of $\mathcal{V}_i$ invariant for $i>0$ and acts trivially on $\mathcal{V}_0$.

<...>
According to Kowalewski, Engel had called his attention to Fano's paper [1896], discussed in Section 3, where Study's complete reducibility theorem for $\mathbf{sl}(2,\mathbb{C})$ was already emplyed to solve a different problem.
<...>
Kowalewski remained in Leipzig as an instructor (Privatdozent) until 1901, when he obtained a position as an asistant professor (ausserordentlicher Professor) at the University at Greifswald, where Study was professor.
<...>
In 1904 Study moved to a professorship at Bonn, and Kowalewski followed him there in 1905. (Engel, who had never held a tenured professorship at Leipzig, became Study's replacement at Greifswald)


I see that the theorem in question is called "Study's complete reducibility theorem". The Google book search for this phrase led to

"Hermann Weyl, 1885-1985: Centenary Lectures" by Chen Ning Yang, Roger Penrose, Armand Borel, Komaravolu Chandrasekharan
Armand Borel: "Hermann Weyl and Lie Groups"

Since E. Study was somewhat of a villain in the 1923 incident related earlier, let me add as a counterpart that he was well aware of this problem around 1890 and had brought the first contribution to it. In fact, S. Lie reports in [L: 785-8] that Study has proven full reducibility (phrased however differently, in terms of projective representations) for $\mathfrac{sl}_2$ in
an unpublished manuscript and that it was quite sure it would be frue more generally for $\mathfrac{sl}_n$. In a letter to S. Lie (December 31, 1890), referred to in [Hw], Study even goes as far as conjecturing it should hold for a simple or semisimple Lie algebras. To both of them, this was an important problem. The manuscript was not published, apparently because the proof appeared too complicated and simplifications were hoped for.


The Ultimate Study Theorem

Let's derive the following equations:
Assume the following equations are true:

Study = Not Fail
Not Study = Fail

Then:
Study + Not Study = Not Fail + Fail
Study (1 + Not) = Fail (1 + Not)

Therefore:
Study = Fail

It was also found as a side effect ...



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Waylon Jennings

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