Abstract
Algebra| Abstract
Algebra |
In General > s.a. elementary (real/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; MacLane & Birkhoff 67; Goldhaber & Ehrlich
70; Kurosh 72; Lang 84; Hazewinkel et al 04 [+ rings and modules].
@ Ug books: Herstein 75.
Algebra > s.a. Character;
Division Algebra; Ideal;
ring.
$ Def: A vector space
(V, +, K) with a multiplication · such
that (V, +, · ) is a ring, and 
(xy)
= (x) y = x(y).
* Result: There are about 1151 consistent algebras in addition to
the reals, which can be formulated by weakening the Field axioms (
200
have
been rigorously proven to be self-consistent).
@ Operations: Borowiec & Marcinek JMP(00) [crossed product].
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 C.
* Result: If A is
a commutative Banach algebra with identity, and C a maximal ideal,
then A/C 
C;
Thus, there is a correspondence between maximal ideals C and kernels
of characters 
;
ker =
C.
Commutative Algebra
@ General references: Zariski & Samuel 58, 60; Bourbaki 62; Hartley & Hawkes
70; Stanley 83; Kunz 97.
@ Differential calculus: Baehr et al JPA(95).
*-Algebra > s.a. C*-algebra.
$ Def: An algebra 
with an involution operation *: → .
* Involution: A map *: → on
an algebra over K,
which is anti-linear, (A + 
B)*
= A* + * B*,
where * is complex conjugation if K = C and
has no
effect if K = 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.
* Moyal *-product
deformation: If is
the algebra Cinfty(R2)
or the real/complex polynomials, we can define the deformed associative product
This is the unique non-trivial deformation of ; Notice
that {F,G}1
is the Poisson bracket.
* Filtered algebra: One
that that be written as A = 
k =
0infty A(k),
with A(k) 
A(k+1);
e.g., observable algebras.
* Jordan algebra: One
in which the product satisfies [a, b, a2]
= 0 for all
a, b, with [a, b, c]:= (ab)c – a(bc)
(the "associator");
This property is weaker than associativity; > s.a. formulations
of quantum mechanics [Jordan
geometry].
* 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 1930's by Gel'fand and others; Examples
are function, operator, and Banach algebras.
@ Topological: Beckenstein, Narici & Suffel 77.
@ Jordan algebra: Raptis mp/01 [Jordan-Lie
superalgebras]; Niestegge IJTP(04) [and quantum observables]; Rios mp/05 [exceptional,
spectrum].
@ Bialgebras: Xu m.QA/00-in
[Gel'fand-Dorfman, rev]; > s.a. lie 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]; > s.a. deformation
quantization [Moyal].
> Related topics:
see Boolean,
clifford, Cuntz, Fréchet,
grassmann, Hecke, Heyting, Hopf, lie,
poisson algebra; [Sigma-Algebra].
Online Resources > see Internet Encyclopedia of Science pages.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
14 jun 2008