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... —$\mathfrak{Joseph}$ :-)
 A: 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..."

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