topic on the algebraic foundations of quantum algebraic topology

topic on the algebraic foundations of quantum algebraic topology

0.1

Topic entry on the algebraic foundations of Quantum Algebraic Topology.

(a). Quantum

Algebraic Topology

(QAT)

is defined as

the mathematical and physical study of general theories of quantum

algebraic structures

from the standpoint of Algebraic Topology,

Category Theory

and their

Non-Abelian

extensions

in

Higher Dimensional Algebra

and

Supercategories

(b). Several suggested new QAT topics are:

1.

Poisson algebras

,

Quantization

methods and Hamiltonian algebroids

2.

K-S

Theorem

and its Quantum

algebraic

consequences

in QAT

3.

Logic Lattice algebras or Many-Valued (MV)

Logic algebras

4.

Quantum MV-Logic

algebras

and

Ł

-

M

n

-

noncommutative lattice

algebras

5.

Quantum Operator Algebras

( such as :

involution

, *-algebras, or

*

-algebras,

von Neumann algebras

,
, JB- and JL- algebras,

C

*

- or C*- algebras,

6.

Quantum von Neumann algebra and subfactors

7.

Kac-Moody and K-algebras

8.

Hopf algebras

,

Quantum Groups

and Quantum group algebras

9.

Quantum Groupoids

and weak Hopf

C

*

-algebras

10.

Groupoid

C*-

Convolution algebras

and *-Convolution Algebroids

11.

Quantum Spacetimes and

Quantum Fundamental Groupoids

12.

Quantum Double

Algebras

13.

Quantum Gravity

,

supersymmetries

, supergravity,

superalgebras

and graded ‘Lie’ algebras

14.

Quantum

Categorical algebra

and Higher Dimensional,

Ł

-

M

n

- Toposes

15.

Quantum

R-categories

, R-supercategories and Symmetry Breaking

16.

Extended Quantum Symmetries in Higher Dimensional Algebras (HDA), such as:

algebroids, double algebroids,

categorical

algebroids,

double groupoids

,

convolution algebroids, groupoid

C

*

-convolution algebroids

17.

Universal algebras

in R-Supercategories

18.

Supercategorical algebras (SA) as concrete

interpretations

of the Theory of

Elementary

Abstract Supercategories (ETAS).

19.

Quantum Non-Abelian Algebraic Topology (QNAAT)

20.

Noncommutative Geometry

,

Quantum Geometry

, and Non-Abelian Quantum

Algebraic Geometry

21.

Other – Miscellaneous

[please add here your

additions

, changes, editing,
remarks, proofs, conjectures, and so on…]

References

1

Alfsen, E.M. and F. W. Schultz:

Geometry of

State Spaces

of Operator Algebras

, Birkäuser, Boston–Basel–Berlin (2003).

2

Atyiah, M.F. 1956. On the

Krull-Schmidt theorem

with applications to sheaves.

Bull. Soc. Math. France

,

84

: 307–317.

3

Auslander, M. 1965. Coherent Functors.

Proc. Conf. Cat. Algebra, La Jolla

,
189–231.

4

Awodey, S. & Butz, C., 2000,

Topological Completeness

for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168–1182.

5

Awodey, S., 1996, ”

Structure

in Mathematics and Logic: A Categorical

Perspective

”, Philosophia Mathematica, 3, 209–237.

6

Awodey, S., 2004, ”An Answer to Hellman’s Question: Does Category Theory Provide a Framework for Mathematical Structuralism”, Philosophia Mathematica, 12, 54–64.

7

Awodey, S., 2006, Category Theory, Oxford: Clarendon Press.

8

Baez, J. & Dolan, J., 1998a, ”Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes”, Advances in Mathematics, 135, 145–206.

9

Baez, J. & Dolan, J., 2001, ”From

Finite Sets

to Feynman Diagrams”, Mathematics Unlimited – 2001 and Beyond, Berlin: Springer, 29–50.

10

Baez, J., 1997, ”An

Introduction

to n-Categories”, Category Theory and Computer Science, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1–33.

11

Baianu, I.C.: 1970,

Organismic Supercategories

: II. On Multistable Systems.

Bulletin of

Mathematical Biophysics

,

32

: 539-561.

12

Baianu, I.C.: 1971b,

Categories

,

Functors

and Quantum

Algebraic
Computations

, in P. Suppes (ed.),

Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science

, September 1–4, 1971, Bucharest.

13

Baianu, I.C. and D. Scripcariu: 1973, On

Adjoint Dynamical Systems

.

Bulletin of Mathematical Biophysics

,

35

(4), 475–486.

14

Baianu, I.C.: 1973, Some Algebraic Properties of

(M,R)

– Systems.

Bulletin of Mathematical Biophysics

35

, 213-217.

15

Baianu, I.C.: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory.

Bulletin of Mathematical Biology

,

39

: 249-258.

16

Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of

Quantum Automata

and N-Valued Łukasiewicz Algebras in

Relation

to Dynamic Bionetworks,

(M,R)

–Systems and Their Higher Dimensional Algebra,

Abstract and Preprint of Report

:

h

t

t

p

:

/

/

w

w

w

.

a

g

.

u

i

u

c

.

e

d

u

/

f

s

401

/

Q

A

u

t

o

.

p

d

f

and

h

t

t

p

:

/

/

w

w

w

.

m

e

d

i

c

a

l

u

p

a

p

e

r

s

.

c

o

m

/

q

u

a

n

t

u

m

+

a

u

t

o

m

a

t

a

+

m

a

t

h

+

c

a

t

e

g

o

r

i

e

s

+

b

a

i

a

n

u

/

17

Baianu, I.C., R. Brown and J.F. Glazebrook. : 2007a, Categorical Ontology of Complex Spacetime Structures: The Emergence of Life and Human Consciousness, Axiomathes, 17: 35-168.

18

Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.

19

Barr, M. and Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag.

20

Barr, M. and Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM.

21

Bell, J. L., 1981, ”Category Theory and the

Foundations of Mathematics

”, British Journal for the Philosophy of Science, 32, 349–358.

22

Bell, J. L., 1982, ”Categories, Toposes and Sets”, Synthese, 51, 3, 293–337.

23

Bell, J. L., 1986, ”From Absolute to Local Mathematics”, Synthese, 69, 3, 409–426.

24

Bell, J. L., 1988, Toposes and Local

Set Theories

: An Introduction, Oxford: Oxford University Press.

25

Birkoff, G. & Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS.

26

Borceux, F.: 1994,

Handbook of Categorical Algebra

, vols: 1–3,
in

Encyclopedia of Mathematics and its Applications

50

to

52

, Cambridge University Press.

27

Bourbaki, N. 1961 and 1964:

Algèbre

commutative

.

,
in Èléments de Mathématique., Chs. 1–6., Hermann: Paris.

28

BJk4)
Brown, R. and G. Janelidze: 2004, Galois theory and a new

homotopy
double groupoid

of a map of spaces,

Applied Categorical
Structures

12

: 63-80.

29

Brown, R., Higgins, P. J. and R. Sivera,: 2007a,

Non-Abelian
Algebraic Topology

, in preparation.

http://www.bangor.ac.uk/ mas010/nonab-a-t.html ;

http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf

30

Brown, R., Glazebrook, J. F. and I.C. Baianu.: 2007b, A Conceptual, Categorical and Higher Dimensional Algebra Framework of

Universal

Ontology and the Theory of Levels for Highly

Complex Structures

and Dynamics.,

Axiomathes

(17): 321–379.

31

Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a

Hausdorff space

.,

Theory and Applications of Categories

10

, 71-93.

32

Brown, R., and Hardy, J.P.L.:1976,

Topological groupoids

I: universal constructions,

Math. Nachr.

, 71: 273-286.

33

Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules,

Cah. Top. Géom. Diff.

17

, 343-362.

34

Brown R and Razak Salleh A (1999) Free crossed

resolutions

of groups and

presentations

of modules of

identities

among relations.

LMS J. Comput. Math.

,

2

: 25–61.

35

Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc.

80

: 1-34.

36

Buchsbaum, D. A.: 1969, A note on

homology

in categories., Ann. of Math.

69

: 66-74.

37

Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes,

Adv. in Math.

179

, 291-317.

38

Bunge, M., 1984, ”Toposes in Logic and Logic in Toposes”, Topoi, 3, no. 1, 13-22.

39

Bunge M, Lack S (2003) Van Kampen theorems for toposes.

Adv Math

,

179

: 291-317.

40

Cartan, H. and Eilenberg, S. 1956.

Homological Algebra

, Princeton Univ. Press: Pinceton.

41

Cohen, P.M. 1965.

Universal Algebra

, Harper and Row: New York, London and Tokyo.

42

Connes A 1994.

Noncommutative geometry

. Academic Press: New York.

43

Croisot, R. and Lesieur, L. 1963.

Algèbre noethérienne non-commutative.

,
Gauthier-Villard: Paris.

44

Baianu, I. C. et al. 2008. Quantum Non-Abelian Algebraic Topology (QNAAT): PM Exposition lec

i

⁢

d

=

68

.

[This is an entry-in-progress for a new topic entry on the Algebraic

Foundations

of Quantum Algebraic Topology.]

Title

topic on the algebraic foundations of quantum algebraic topology

Canonical name

TopicOnTheAlgebraicFoundationsOfQuantumAlgebraicTopology

Date of creation

2013-03-22 18:14:21

Last modified on

2013-03-22 18:14:21

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

34

Author

bci1 (20947)

Entry type

Topic

Classification

msc 08A99

Classification

msc 08A70

Classification

msc 18E05

Classification

msc 18-00

Classification

msc 55U40

Classification

msc 55-02

Classification

msc 81Q60

Classification

msc 08A05

Classification

msc 81R15

Synonym

Algebraic Foundations of Quantum Algebraic Topology topic

Related topic

QuantumAlgebraicTopology

Related topic

NonAbelianTheories

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlgebras

Generated on Fri Feb 9 14:05:23 2018 by

LaTeXML