# Fraktur symbols for Lie algebras

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... —$\mathfrak{Joseph}$ :-)

• 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
• When I was an undergraduate at Columbia, a student questioned Lang about some written specimen of Fraktur (probably $\mathfrak p$), and he said, if I recall correctly, “You don’t know the German alphabet?” and proceeded to write out $\mathfrak {a,b,c}$ etc. – Lubin Oct 13 '17 at 19:57
• Unless I'm mistaken, it's pretty commonplace in set theory and cardinal arithmetic to use $\mathfrak{m},\mathfrak{n},\dots$ for arbitrary (possibly infinite) cardinal numbers. This goes back to at least Sierpinski in 1956, and I believe it goes all the way back to Cantor in the 1800's. – Alec Rhea Oct 14 '17 at 3:13

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 $\mathfrak{G}$ was commonly used to denote a group, while the ordinary italic or 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 early 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 (in Fraktur $\mathfrak{B}$ and $\mathfrak{V}$, 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 lower case letters like $m, p, q$ ($\mathfrak{m}, \mathfrak{p}, \mathfrak{q}$ in Fraktur), which most people find impossible to imitate by hand. But for Hilbert and others this was quite natural notation, as was lower case $\mathfrak{f}$ for the German word 'Fuhrer' (printed with Umlaut over 'u'), now usually called the "conductor".

By the way, in Lie algebra theory the lower case letter $g$ (or $\mathfrak{g}$) was naturally used because the Lie algebra was first regarded as an infinitesimal group.

• 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.

• :-) $\mbox{} \mbox{}$ – Joseph O'Rourke Feb 6 '12 at 1:45
• Best MO anecdote EVAR. – Mariano Suárez-Álvarez 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
• To me $\frak{A}$ evocates the number $21$ . – Sylvain JULIEN Oct 13 '17 at 21:06

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.

This may be relevant re Jim Humphreys' reminisces: From a Cornell page on German script, here is Fraktur LaTeX letters compared to handwritten versions:

Another source: Wikipedia article. Note:

"The word Fraktur is derived from the Latin fractus meaning "broken..."

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

A subsidiary point is that, whether or not one uses typeset fraktur of some sort, the handwritten version cannot possibly truly imitate the print, any more than cursive writing (e.g., in Western Europe) imitates print. Sure, some resemblances, but not entirely obvious. To my perception, this is very obvious in the case of Russian in Cyrillic versus handwritten Cyrillic, the latter being unintelligible to me.

Similarly, there were various handwriting systems introduced in Germany in the 19th century, and I think the one that survived (at least up to a point) in the U.S., due to emigres such as Artin and Siegel at Princeton, was approximately "Sutterlin" (google-able), which, without naming it, was implicitly promoted decades ago at Princeton as a way to hand-write fraktur. (Despite many good-intentioned attempts to recreate the blocky-spikey fraktur in handwriting, no one in Germany did that!) A form of that is what I promote to my grad students nowadays, too: not to attempt to create literal fraktur of any sort by hand, but to write the Sutterlin conception of handwritten versions.

There were other versions of handwritten "fraktur" suggested, too, but none of them attempted to literally recreate the printed forms!!!

• Sütterlin was an early twentieth century version of the standard Kurrentschrift (or simply Kurrent) that had been in use for centuries [see the Wikipedia article]. Both are essentially unreadable (now), unless written carefully. – David Handelman Oct 13 '17 at 23:31
• @DavidHandelman, I am by no means an expert on this, but for some reason I had the impression that there were several somewhat-competing conceptions of handwriting corresponding to "fraktur"... so that "Sutterlin" would have been unintelligible to other branches of the handwriting tree there. Can you comment? – paul garrett Oct 14 '17 at 0:07

Let me point out a counter-example, in which Fraktur is not used for Lie-algebras.

It's Непрерывные группы (literally "Continuous Groups") by Lev Pontryagin, published in 1946 in the USSR. (Sorry, in fact, I have a Japanese translation from the second Russian edition in 1954. It is "連続群論" in 1957. See Worldcat entry for the bibliography.) It uses Fraktur for classical Lie groups(!), and Roman for their Lie algebras.

See the middle of p. 521, the left photo. The word "リー群" means "Lie group(s)", as you can see from the Japanese Wikipedia page. At the bottom of the same page, it says "$\mathfrak{H}_r$ のリー代数 $H_r$" ($\mathfrak{H}_r$'s Lie algebra $H_r$). See also p. 545, the right photo. You can see the Dynkin diagrams of the classical and exceptional Lie algebras.

Click to enlarge. These photo citations must be ok under the copyright law.

(I originally posted this as a MSE question. I think these photo citations are allowed under the copyright )