Abstract Algebra| Abstract Algebra |
In General > s.a. elementary (real and complex) algebra;
Homological Algebra.
* Idea: The study of properties of sets
A with some operations, internal or external (defined with a field K).
@ General references: Van der Waerden 31;
Bourbaki 42–62;
Jacobson 51–64;
Birkhoff & MacLane 53;
Chevalley 56;
Kurosh 65;
Mac Lane & Birkhoff 67;
Goldhaber & Ehrlich 70;
Kurosh 72;
Lang 84;
Fan et al 99;
Hazewinkel et al 04 [+ rings and modules];
Knapp 06;
Eie & Chang 10;
Adhikari & Adhikari 14 [IIb];
Reis & Rankin 16;
Lawrence & Zorzitto 21 [intro].
@ Undergraduate books: Herstein 75.
Algebra > s.a. Character; Coalgebra;
Division Algebra; Ideal; ring.
$ Def: A vector space
(V, +, K) with a multiplication · such that
(V, +, · ) is a ring, and α(xy)
= (αx) y = x(αy) for all
α ∈ K and x, y ∈ V.
* Result: There are about 1151
consistent algebras in addition to the reals, which can be formulated by
weakening the Field axioms (more than 200 have been rigorously proven to
be self-consistent).
@ Operations:
Borowiec & Marcinek JMP(00) [crossed product].
> Examples:
see quaternions.
Banach Algebra
$ Def: A complete normed algebra.
* Result: (Gel'fand)
If every non-zero a in a commutative Banach algebra with
identity A is invertible, A is isomorphic to \(\mathbb C\).
* Result: If A is
a commutative Banach algebra with identity, and C a maximal ideal,
then A/C ≅ \(\mathbb C\);
Thus, there is a correspondence between maximal ideals C and kernels
of characters χ; ker χ = C.
Commutative Algebra
@ General references:
Zariski & Samuel 58, 60;
Bourbaki 62;
Hartley & Hawkes 70; Stanley 83;
Kunz 85;
Li 04;
Patil & Storch 10 [and algebraic geometry];
Singh 11.
@ Differential calculus: Baehr et al JPA(95).
*-Algebra > s.a. C*-algebra.
$ Def: An algebra \(\cal A\)
with an involution operation *:
\(\cal A\) → \(\cal A\).
* Involution: A map
*: \(\cal A\) → \(\cal A\) on an
algebra over K, which is anti-linear, (A + λB)*
= A* + λ* B*, where * is complex conjugation if
K = \(\mathbb C\) and has no effect if K = \(\mathbb R\), and
satisfies (AB)* = B*A*, (A*)* = A.
@ References: Bagarello JMP(08) [O*-algebras and quantum dynamics].
Other Algebras and Generalizations
> s.a. deformation quantization; observable algebras.
* Moyal star-product deformation: If \(\cal A\) is
the algebra C∞(\(\mathbb R\)2)
or the real/complex polynomials, we can define the deformed associative product
\[ (F \star_\hbar G)(p,q):= FG(p,q) + \sum_{k\ge1} {({\rm i}\hbar)^k\over2^k k!}\,\{F,G\}_k(q,p) \;,\]
\[{\rm where}\quad \{F,G\}_k:= \sum_{i=0}^k (-1)^i {k\choose i}\,
{\partial^kF\over\partial^{k-i}p\,\partial^iq}\,{\partial^kG\over\partial^ip\,\partial^{k-i}q}\;.\]
This is the unique non-trivial deformation of \(\cal A\); Notice
that {F,G}1
is the Poisson bracket.
* Filtered algebra: One
that that be written as A = ∪k
= 0∞
A(k),
with A(k)
⊂ A(k+1);
e.g., observable algebras.
* Malcev algebra: A (non-associative)
algebra in which the product satisfies x2
= 0 for all x and J(x, y, xz)
= J(x, y, z) x for all x,
y, z, where J(x, y, z):=
(xy) z + (yz) x + (zx) y
[the Jacobi-type combination of three elements]; Example:
All Lie algebras, in which J(x, y, z)
= 0 for all x, y, z.
* Topological algebra: An
algebra with a suitably related (quasi)topological structure; The theory
was developed in the late 1930s by Gel'fand and others; Examples are
function, operator, and Banach algebras.
@ Topological: Beckenstein, Narici & Suffel 77.
@ Bialgebras:
Xu m.QA/00-conf [Gel'fand-Dorfman, rev];
Rezaei-Aghdam et al a1401 [Leibniz bialgebras];
> s.a. lie algebras.
@ n-ary algebras:
Goze et al JAPM-a0909;
Fairlie & Nuyts JPA(10)-a1007 [conditions for ternary algebras].
@ Related topics: in Jordan in(72) [Malcev];
Jaganathan mp/00 [quantum, intro];
Bandelloni & Lazzarini NPB(01)ht/00 [W3];
Gudder & Greechie IJTP(05) [sequential effect algebras];
Doubek et al a0705-ln [deformation theory];
Huang et al IJTP(10) [causal algebras].
Examples: see Boolean,
clifford, Cluster,
Cuntz, Fréchet,
Galois, grassmann, Hecke,
Heyting, Hopf, lie,
poisson algebra; deformation quantization [Moyal algebra];
[Sigma-Algebra]; Simple Algebra;
Tangle [diassociative algebras]; Temperley-Lieb Algebra.
Online Resources > see Internet Encyclopedia of Science pages.
main page
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