Group Theory| Group Theory |
In General
> s.a. types of groups [including generalizations].
* History:
The theory was invented by Galois in the early XIX century.
$ Def: A group is a
pair (G, \(\circ\)), with G a set and \(\circ\) an
operation \(\circ\) : G × G → G,
which is associative, has a neutral element, and such that each element
has an inverse; I.e., a group is a monoid with inverse.
* Specifying a group:
For finite groups, one can give the full multiplication table;
In general, one gives a list of elements and a composition law in some
representation, or a group presentation; The fundamental problem of group
theory is the isomorphism problem, that of deciding whether
any two finite presentations correspond to isomorphic groups; It was
proved unsolvable in the late 1950s;
> s.a. Combinatorial Group Theory;
Group Presentation; Isomorphism;
representations.
* Main branches:
Abstract group theory; representation theory.
@ II: Falicov 66;
Leech & Newman 69;
Armstrong 88;
Moody 94;
Mirman 95 [II/III];
Snaith 03 [and rings, etc];
Ronan 06 [esp Monster group];
Reis 11.
@ III: Weyl 46;
Kurosh 55;
Hall 59;
Hamermesh 62;
Scott 64;
Rotman 65;
Suzuki 82;
Zassenhaus 99;
Roman 12;
Machì 12 [many exercises].
@ Geometric group theory:
de la Harpe 00;
Chiswell in Bullett et al 17.
@ Books, special emphasis:
Babakhanian 72 [cohomological methods];
Asche 89 [with software];
Geoghegan 08 [topological methods];
Steeb et al 12 [problems and solutions].
@ Related topics: Stillwell BAMS(82) [isomorphism problem].
> Applications: see algebraic topology;
ordinary and partial differential equations;
Group Theory in Physics below.
> Online resources:
see Wikipedia page.
Related Concepts and Structure
> s.a. Center; Coset; group
action on a set; lie group [metric]; measure;
Rank; Word.
* Conjugate elements: Two elements
g1, g2
in G (two subgroups H1,
H2 of G) are conjugate if there exists
a g in G such that g1 = g
g2 g−1
(H1 = g H2
g−1); This is an equivalence relation.
* Conjugacy Classes:
* Subgroups: Only if N
is a normal subgroup is G/N a group; The order of a subgroup
must be a divisor of the order of the group (think about cosets).
* Sylvester graph of a group:
> see, e.g., matter.
@ References: Balantekin AIP(10)-a1011 [character expansions of products of invariant functions on G].
Operations on Groups > s.a. lie
algebras [group contractions]; Semidirect Product.
* Extension of a group A by a
group C: A new group B ⊃ A with B/A
= C; Expressed by the exact sequence
0 → A → B → C → 0 .
* Free product:
Subgroups > s.a. group actions [stabilizer];
Little Group; Torsion Subgroup.
* Normal: (H \(\triangleleft\) G)
An H such that for all g in G,
g−1Hg = H;
Alternatively, H is the kernel of some homomorphism.
* Normalizer of a subset S:
The subgroup of G defined by NG(S):=
{g ∈ G | g−1Sg = S}
≡ {g ∈ G | Sg = gS}, i.e., the biggest subgroup of G
in which S is normal; If S is a subgroup, S \(\triangleleft\) N(S),
and S \(\triangleleft\) G is equivalent to N(S) = G;
Special cases: If S = {s}, one element, N(S)
= C(S), but in general C(S) ⊂ N(S).
* Commutant: The normal subgroup Q
of elements of the form q = q1
q2 ... qm,
where qi
= gg'g−1g'−1;
Remark: The quotient group G/Q is Abelian.
* Centralizer of a subset
S: The subgroup of G defined by
CG(S):= {g ∈ G | g−1sg = s , ∀s ∈ S} ≡ {g ∈ G | gs = sg , ∀g ∈ S} .
And Physics > s.a. gauge theory; general
relativity; lie groups; quantum theory
and canonical approach; symmetries.
* Idea: It is a very effective way
of formulating and exploiting the symmetries of a system; The knowledge of a symmetry
group gives the level scheme, branching ratios and selection rules; Symmetries are
usually broken, but the approximation is still useful.
* History: Symmetry in physics was
introduced with special relativity and the Lorentz group; Noether's theorem; 1932,
Heisenberg joins p and n into an SU(2) doublet and postulates an
SU(2)-invariant Lagrangian; 1954, local symmetries introduced with Yang-Mills theory.
@ Books and overviews: Kahan 65;
Wybourne 74;
Balachandran & Trahern 84;
Cornwell 84;
Tung 85 (and
Aivazis 91);
Isham 89;
Sternberg 91;
Fässler & Stifel 92;
Ludwig & Falter 96;
Teodorescu & Nicorovici 04;
Ma & Gu 04 [problems and solutions];
Bonolis RNC(04) [history];
Dresselhaus et al 08 [in condensed matter, r PT(08)nov];
Ramond 10;
Balachandran et al 10 [and Hopf algebras];
Jeevanjee 11 [r PT(12)apr];
Belhaj a1205-ln;
Coddens 15 [spin and quantum mechanics];
Zee 16 [in a nutshell];
Martin-Dussaud GRG(19)-a1902 [lqg and spin foams];
Ma 19;
Wilson a2009 [group-theorist's perspective].
@ For particle physics:
Barnes 10 [standard model];
Haywood 10 [symmetries and conservation laws];
Costa & Fogli 12;
Saleem & Rafique 12.
@ And dynamics:
Aldaya & Azcárraga FdP(87).
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