# Fraktur symbols for Lie algebras: $\mathfrak{g}$, etc.

Does anyone know when and why the Fraktur script was introduced for Lie and other algebras—$\mathfrak{g}$, $\mathfrak{gl}_n$, $X/\mathfrak{g}$, $\mathfrak{g}\oplus\mathfrak{g}$, $\mathfrak{su}$, $\mathfrak{M}_g$, etc.? And introduced by whom? Is its use pretty much restricted to algebra, or was it in the past employed in, say, geometry as well, but has only survived to the current time within algebra? (Or maybe it is currently used outside of algebra and I am just ignorant of those areas.)

The typeface itself goes back to the 15th century. The generally illuminating website "Earliest Uses of Various Mathematical Symbols" seems not to shed light on this issue.

I find the Fraktur font adds a certain elegance and mystery to the subjects that utilize it! I'm a bit envious, not working in those fields... —$\mathfrak{Joseph}$ :-)

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Various German algebraists (Noether for example) used Fraktur script in their work at least back to the 1910s. So one reasonable guess would be Hermann Weyl, who advanced the theory of Lie algebras in the 1930s. –  Eric Rowell Feb 6 '12 at 2:20
Reminds me of johnlangdon.net/angelsanddemons.php if you click on any of the "ambigrams" it rotates it 180 degrees. In response to your question, I think unreadable letters are introduced out of malice, to make life difficult for other people. In the other direction, the Korean alphabet was commissioned (work done by a group of scholars) by King Sejong, to help a largely illiterate populace who did not speak (or write) Chinese, when learned discourse was written in Chinese. Sort of like Latin. –  Will Jagy Feb 6 '12 at 4:26
Will, I remember my algebraic number theory class, where the Fraktur letters were written with such elegance that I couldn't tell a "$\mathfrak p$" from a "$\mathfrak q$". I guess it did add to the mystery :) –  B R Feb 6 '12 at 5:47
@BR, I took German in school, at some point my mother got an inexpensive box of books at an auction, so I had a dozen books in Fraktur. I couldn't make heads or tails of them. –  Will Jagy Feb 6 '12 at 6:34

Some of what's been said so far about the history makes good sense, but by no means all. Let me add my own perspective, for what it's worth. The font called Fraktur by LaTeX (also known as "gothic") was widely used historically in German printing (though I don't own a Gutenberg Bible). It naturally crept into mathematical usage and notation. For instance, the upper case Fraktur letter G was commonly used to denote a group, while the ordinary roman G denoted an element of the group.

This convention persisted among emigres like Walter Feit who grew up in Vienna (and escaped on the last children's train though his parents didn't). In his course at Yale which I took as a graduate student he filled the blackboard elegantly with ornate symbols, which I sort of learned to copy down (see his Benjamin lecture notes on character theory from that era). But I had actually encountered Fraktur when I first learned some German grammar in high school. It was a mediocre working class public school but located among various ethnic enclaves (including Italian and German), so those languages got taught for a while in two year sequences. The principal wouldn't let me and a classmate of German descent start with the second course, so we sat in the back of the classroom in the first year course and worked ahead on our own. The old German textbooks available in that postwar era were all in Fraktur, which had been promoted during the Third Reich as the "correct" way to print the language of the master race. So I did learn to distinguish upper case B and V, etc.

The point is that group theory and Lie groups in particular were actively developed by German mathematicians in the nineteenth century; they were not inventing exotic notation when they used these particular letters as symbols. In number theory there is still a widespread tendency to use even the lower case letters like m in Fraktur, which most people find impossible to imitate by hand. But for Hilbert and others this was quite natural notation, as was the lower case Fraktur f (for Fuhrer, now usually called "conductor").

By the way, in Lie algebra theory the lower case letter was naturally used because the Lie algebra was first regarded as an infinitesimal group.

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I want to upvote this more than once. –  Ketil Tveiten Mar 2 '12 at 12:41
Incidentally, the wartime German mathematical journal Deutsche Mathematik used the Roman font for running text in the first few issues, but soon switched over to black letter ("Fraktur"), presumably because it was thought to be more Teutonic. I don't know whether they had the time to switch back to Roman before closing down definitively; the library in my current place of work doesn't have copies of the journal. –  Chandan Singh Dalawat Mar 2 '12 at 14:41
May I point out a mistake: During the third Reich the Nazi tried to abolish Fraktur. In fact they succeeded. I do not know why today Fraktur is associated with Nazi - historically this is false. I have read many books in fraktur and like it. My grandmother' handwriting was "Kurrent" which was more beautiful and easier to read that todays handwriting. The aim was to minimize hand movements in directions other than the main direction (NE-SW). In very old manuscripts, Fraktur was always the type for German texts, and Latin type was the type for Latin texts. You find texts containing both. –  Peter Michor Jan 1 '13 at 20:11
@Peter: My sources are mainly anecdotal, so I can't disagree with your version of the history which seems to be acquired more directly. Growing up as I did among so many people with Italian or German heritage, I never learned any really impartial version of early 20th century history. It's certainly possible that the textbooks in Fraktur which I used in high school were made obsolete by the Nazis, but for some reason I grew up thinking the opposite. Our public schools never taught any history after 1920, since FDR too was a toxic subject. –  Jim Humphreys Jan 3 '13 at 23:50
@Jim: The german Wikipedia <A HREF="de.wikipedia.org/wiki/…; has a good historical description. If you switch to English, the historical description becomes much shorter. I was not completely correct: In the beginning the Nazi favored Fraktur a little (out of romantic feeling), but there was a sudden change against Fraktur in 1941. –  Peter Michor Jan 9 '13 at 8:13

I don't know the mathematical history of Fraktur, but the following story (I'm not sure whether it's true but it's at least imaginable, and I didn't make it up) might make you feel better about working in a Fraktur-less field. The Detroit Free Press (one of Detroit's two major daily newspapers) had its name in very large Fraktur type in its masthead. It took a long time (years, not days) before someone pointed out that it said "Vetroit", not "Detroit".

I can vouch for a similar confusion on the basis of my own experience. In mathematical logic, we often use Fraktur capital letters for models and the corresponding ordinary (italic) letters for the underlying sets of the models. Far too many students assume that the Fraktur A ($\mathfrak A$) is intended to be a U.

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:-) $\mbox{} \mbox{}$ –  Joseph O'Rourke Feb 6 '12 at 1:45
Best MO anecdote EVAR. –  Mariano Suárez-Alvarez Feb 6 '12 at 2:08
@Mariano, shouldn't it be EDER? Which, as I had to look up, is a river. In Germany. So it all fits. –  Will Jagy Feb 6 '12 at 4:49
–  Frank Thorne Mar 2 '12 at 2:22

I am pretty sure that the first use of Fraktur by Sophus Lie occurs in 1869, which is before he invented Lie groups or Lie algebras. It appears in his paper Repraesentation der Imaginaeren der Plangeometrie, in the first volume of his collected works, to represent the plane. I assume that it was standard practice in German mathematics to use German script letters, because they were used in number theory by Dirichlet and others before 1869. Lie groups slowly evolved in the mid to late 1870's, entering their final form in the 1880s. But in discussing Lie algebras, he rarely uses Fraktur fonts. He usually talks about a group G and then writes out its Lie algebra. I didn't run into Fraktur fonts before 1891, Die linearen homogenen gewohnlichen Differentialgleichungen, used to describe a sort of generating function for a Lie algebra. Maybe an expert (Thomas Hawkins or Peter Olver) would have better luck.

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Fraktur was the standard font for cardinal numbers, for example, in writing the continuum hypothesis as $\aleph_1=\frak c$.

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