Quick Calculation 15.1.2

A First Course in String Theory


Recall that a group is a set which is closed under an associative multiplication; it contains an identity element, and each element has a multiplicative inverse. Verify that \displaystyle{U(1)} and \displaystyle{U(N)}, as described above, are groups.



A group is a set, G, together with an operation \displaystyle{\bullet} (called the group law of G) that combines any two elements a and b to form another element, denoted \displaystyle{a \bullet b} or \displaystyle{ab}. To qualify as a group, the set and operation, \displaystyle{(G, \bullet)}, must satisfy four requirements known as the group axioms:


For all a, b in G, the result of the operation, \displaystyle{a \bullet b}, is also in G.


For all a, b and c in G, \displaystyle{(a \bullet b) \bullet c = a \bullet (b \bullet c)}.

Identity element

There exists an element e in G such that, for every element a in G, the equation \displaystyle{e \bullet a = a \bullet e = a} holds. Such an element is unique, and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G, commonly denoted \displaystyle{a^{-1}} (or \displaystyle{-a}, if the operation is denoted “+”), such that \displaystyle{a \bullet b = b \bullet a = e}, where e is the identity element.

— Wikipedia on Group (mathematics)


The axioms for a group are short and natural… Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.

— Richard Borcherds in Mathematicians: An Outer View of the Inner World



2019.07.28 Sunday ACHK

Alfred Tarski, 3

The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.

— Wikipedia on Tarski’s undefinability theorem


Tarski’s 1969 “Truth and proof” considered both Gödel’s incompleteness theorems and Tarski’s undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.

— Wikipedia on Alfred Tarski



2019.07.20 Saturday ACHK

PhD, 3.7.2

碩士 4.7.2 | On Keeping Your Soul,






  1. 如果你是自資,就即是不拿學校的資助。那樣,你就不是僱員。研究以外的工作,例如做助教等,可以一概不理。

  2. 同理,你的博士導師再不會是,你工作上的上司。反而,你是消費者,他是你的僱員。













— Me@2019-07-06 10:57:22 PM



2019.07.10 Wednesday (c) All rights reserved by ACHK