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.

@ __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 *x*^{2}
= 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.

@ __Bialgebra__s:
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].

@

**Online Resources**
> see Internet Encyclopedia of Science
pages.

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