This paper gives a brief survey of certain recently developing aspects of the study of loops and quasigroups, focussing on some of the areas that appear to exhibit the best prospects for subsequent research and for applications both inside and outside mathematics.
@article{119175,
author = {Jonathan D. H. Smith},
title = {Loops and quasigroups: Aspects of current work and prospects for the future},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {415-427},
zbl = {1038.20055},
mrnumber = {1780883},
language = {en},
url = {http://dml.mathdoc.fr/item/119175}
}
Smith, Jonathan D. H. Loops and quasigroups: Aspects of current work and prospects for the future. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 415-427. http://gdmltest.u-ga.fr/item/119175/
On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960), 20-104. (1960) | MR 0141119 | Zbl 0096.17404
Vector fields on spheres, Ann. of Math. 75 (1962), 603-632. (1962) | MR 0139178 | Zbl 0112.38102
Geometry and Algebra of Multidimensional Three-Webs, Kluwer Dordrecht (1992). (1992)
Studies in Modern Algebra, Prentice-Hall Englewood Cliffs, NJ (1963). (1963) | MR 0146228 | Zbl 0192.00102
Sporadic Groups, Cambridge University Press Cambridge (1994). (1994) | MR 1269103 | Zbl 0804.20011
Nets and groups I, Trans. Amer. Math. Soc. 46 (1939). (1939) | MR 0000035
Algebraic Combinatorics I, Benjamin/Cummings Menlo Park, CA (1984). (1984) | MR 0882540 | Zbl 0555.05019
Fractals Everywhere, Academic Press San Diego, CA (1988). (1988) | MR 0977274 | Zbl 0691.58001
A Survey of Binary Systems, Springer Berlin (1958). (1958) | MR 0093552 | Zbl 0081.01704
A Guide to Quantum Groups, Cambridge University Press Cambridge (1994). (1994) | MR 1300632 | Zbl 0839.17009
Quasigroups and Loops: Theory and Applications, Heldermann Berlin (1990). (1990) | MR 1125806 | Zbl 0719.20036
An enumeration of knots and links, in ``Computational Problems in Abstract Algebra'', (ed. J. Leech), Pergamon Press, 1969, pp.329-358. | MR 0258014 | Zbl 0202.54703
Split extensions and representations of Moufang loops, Comm. Algebra 23 (1995), 4245-4255. (1995) | MR 1351131 | Zbl 0834.20067
Multiplication groups of quasigroups and loops I, Acta Univ. Carolin. Math. Phys. 34 (1993), 85-99. (1993) | MR 1282769
Multiplication groups of quasigroups and loops II, Acta Univ. Carolin. Math. Phys. 35 (1994), 9-29. (1994) | MR 1366489
Algebraic cohomology groups and loops, Duke Math. J. 14 (1947), 435-463. (1947) | MR 0021939
A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239-246. (1985) | MR 0776477 | Zbl 0572.57002
Gyrosemidirect product structure of bounded symmetric domains, Res. Math. 26 (1994),28-31. (1994) | MR 1290679 | Zbl 0841.20062
Representations of right quasigroups, in ``Proceedings of the Second Asian Mathematical Conference 1995 (Nakhon Ratchasima)'', World Sci. Publishing, River Edge, NJ, 1998, pp.128-131. | MR 1660558 | Zbl 0951.20051
Theory of Multicodimensional $(n+1)$-Webs, Kluwer Dordrecht (1988). (1988) | MR 0998774 | Zbl 0668.53001
The homotopies of quasigroups and universal algebras, in ``Universal Algebra and Quasigroup Theory'', (eds. A. Romanowska and J.D.H. Smith), Heldermann, Berlin, 1992, pp.89-99. | MR 1191228 | Zbl 0818.20088
Logarithms, syndrome functions, and the information rate of greedy loop transversal codes, J. of Comb. Math. and Comb. Comp. 22 (1996), 33-49. (1996) | MR 1418056
Greedy loop transversal codes, metrics, and lexicodes, J. of Comb. Math. and Comb. Comp. 22 (1996), 143-155. (1996) | MR 1418067 | Zbl 0864.94033
Varietal generalizations of Schur multipliers, stem extensions and stem covers, J. Reine angew. Math. 270 (1974),169-183. (1974) | MR 0396786
Some historical aspects of the representation theory of groups and its extension to quasigroups, in ``Universal Algebra and Quasigroup Theory'', (eds. A. Romanowska and J.D.H. Smith), Heldermann, Berlin, 1992, pp.101-117. | MR 1191229 | Zbl 0771.20001
Loop cohomology, Czech. Math. J. 40 (1990),182-194. (1990) | MR 1046286 | Zbl 0728.20057
Characters of finite quasigroups, European J. Combin. 5 (1984), 43-50. (1984) | MR 0746044 | Zbl 0537.20042
Characters of finite quasigroups II: induced characters, European J. Combin. 7 (1986), 131-137. (1986) | MR 0856325 | Zbl 0599.20110
Characters of finite quasigroups III: quotients and fusion, European J. Combin. 10 (1989), 47-56. (1989) | MR 0977179 | Zbl 0667.20053
Characters of finite quasigroups IV: products and superschemes, European J. Combin. 10 (1989), 257-263. (1989) | MR 1029172 | Zbl 0669.20053
Characters of finite quasigroups V: linear characters, European J. Combin. 10 (1989), 449-456. (1989) | MR 1014553 | Zbl 0679.20059
Characters of finite quasigroups VI: critical examples and doubletons, European J. Combin. 11 (1990), 267-275. (1990) | MR 1059557 | Zbl 0704.20056
Stone Spaces, Cambridge University Press Cambridge (1982). (1982) | MR 0698074 | Zbl 0499.54001
A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), 37-65. (1982) | MR 0638121 | Zbl 0474.57003
Groups with an involutory antiautomorphism and K-loops, Res. Math. 23 (1993), 338-354. (1993) | MR 1215219 | Zbl 0788.20034
On loops with cyclic inner mapping groups, Arch. Math. 60 (1993), 233-236. (1993) | MR 1201636
Connected transversals to subnormal subgroups, Comment. Math. Univ. Carolinae 38 (1997), 223-230. (1997) | MR 1455488 | Zbl 0889.20020
Collected Works of Michihiko Kikkawa, Shimane University Matsue, Shimane (1998). (1998)
Inner mappings of Bruck loops, Math. Proc. Camb. Phil. Soc. 123 (1998), 53-57. (1998) | MR 1474864 | Zbl 0895.20052
On K-loops of finite order, Res. Math. 25 (1994), 79-102. (1994) | MR 1262088 | Zbl 0803.20052
Baer-invariants, isologism, varietal laws and homology, Acta Math. 137 (1976). (1976) | MR 0435250 | Zbl 0364.20036
Conformal field algebras with quantum symmetry from the theory of superselection sectors, Comm. Math. Phys. 134 (1990), 139-196. (1990) | MR 1079804 | Zbl 0715.17028
Distributive groupoids in the theory of knots (Russian), Mat. Sb. 119 (1982), 78-88. (1982) | MR 0672410
Algorithmic and Computer Methods in Three-Dimensional Topology (Russian), Moscow University Press Moscow (1991), English translation by M. Hazewinkel and M. Tsaplina: {Algorithmic and Computer Methods for Three-Manifolds}, Kluwer, Dordrecht, 1997. (1991) | MR 1162113
Loops as invariant sections in groups, and their geometry, Canad. J. Math. 46 (1994), 1027-1056. (1994) | MR 1295130 | Zbl 0814.20055
On loops which have dihedral $2$-groups as inner mapping groups, Bull. Austr. Math. Soc. 52 (1995), 153-160. (1995) | MR 1344268 | Zbl 0838.20080
Transversals, commutators and solvability in finite groups, Boll. Un. Mat. Ital. A(7) 9 (1995), 203-208. (1995) | MR 1324621 | Zbl 0837.20026
On the structure of the inner mapping groups of loops, Comm. Algebra 24 (1996),135-142. (1996) | MR 1370527 | Zbl 0853.20049
On connected transversals to subgroups whose order is a product of two primes, European J. Combin. 18 (1997), 915-919. (1997) | MR 1485376 | Zbl 0889.20044
On multiplication groups of loops, J. Algebra 135 (1990), 112-122. (1990) | MR 1076080 | Zbl 0706.20046
On connected transversals to abelian subgroups in finite groups, Bull. London Math. Soc. 24 (1992), 343-346. (1992) | MR 1165376 | Zbl 0793.20064
On connected transversals to abelian subgroups, Bull. Austral. Math. Soc. 49 (1994), 121-128. (1994) | MR 1262682 | Zbl 0799.20020
On connected transversals in infinite groups, Math. Scand. 70 (1992), 172-176. (1992) | MR 1189971 | Zbl 0774.20020
On connected transversals in the projective special linear group, J. Algebra 166 (1994), 455-460. (1994) | MR 1280587 | Zbl 0812.20057
Equivalence, Invariants, and Symmetry, Cambridge University Press Cambridge (1995). (1995) | MR 1337276 | Zbl 0837.58001
A generalised inverse for matrices, Proc. Camb. Phil. Soc. 51 (1955), 406-413. (1955) | MR 0069793
Quasigroups and Loops: Introduction, Heldermann Berlin (1990). (1990) | MR 1125767 | Zbl 0715.20043
Quasiprimitivity and quasigroups, Bull. Austral. Math. Soc. 59 (1999), 473-475. (1999) | MR 1697989 | Zbl 0944.20058
Semilattice-based dualities, Studia Logica 56 (1996), 225-261. (1996) | MR 1382175 | Zbl 0854.08007
The geometry of loops, Math. Notes 12 (1972), 799-805. (1972) | MR 0340461
Quasigroups, geometry and nonlinear geometric algebra, Acta Appl. Math. 50 (1998), 45-66. (1998) | MR 1608635 | Zbl 0903.20038
Mal'cev Varieties, Springer Berlin (1976). (1976) | MR 0432511 | Zbl 0344.08002
Representation Theory of Infinite Groups and Finite Quasigroups, Les Presses de l'Université de Montréal Montreal (1986). (1986) | MR 0859373 | Zbl 0609.20042
Quasigroups, association schemes, and Laplace operators on almost-periodic functions, in ``Algebraic, Extremal and Metric Combinatorics 1986'', (M.-M. Deza, P. Frankl and I.G. Rosenberg, Eds.), Cambridge University Press, Cambridge, 1988, pp.205-218. | MR 1052668 | Zbl 0671.43009
Entropy, character theory and centrality of finite quasigroups, Math. Proc. Camb. Phil. Soc. 108 (1990), 435-443. (1990) | MR 1068446 | Zbl 0712.20042
Combinatorial characters of quasigroups, in ``Coding Theory and Design Theory Part I: Coding Theory'', (ed. D. Ray-Chaudhuri), Springer, New York, NY, 1990, pp.163-187. | MR 1047879 | Zbl 0708.20021
Skein polynomials and entropic right quasigroups, Demonstratio Math. 24 (1991), 241-246. (1991) | MR 1142892 | Zbl 0789.20081
Loop transversals to linear codes, J. of Comb., Info. and System Sci. 17 (1992), 1-8. (1992) | MR 1216961
Quasigroups and quandles, Discrete Mathematics 109 (1992),277-282. (1992) | MR 1192389 | Zbl 0787.20041
A left loop on the $15$-sphere, J. Algebra 176 (1995),128-138. (1995) | MR 1345298 | Zbl 0841.17004
Homotopy and semisymmetry of quasigroups, Algebra Universalis 38 (1997), 175-184. (1997) | MR 1608968 | Zbl 0906.20046
Quasigroup actions: Markov chains, pseudoinverses, and linear representations, Southeast Asian Bull. Math. 23 (1999), 1-11. (1999) | MR 1810836 | Zbl 0944.20059
Post-Modern Algebra, Wiley New York, NY (1999). (1999) | MR 1673047 | Zbl 0946.00001
Abstract space-times and their Lorentz groups, J. Math. Phys. 37 (1996). (1996) | MR 1390256 | Zbl 0863.20035
Thomas precession and its associated grouplike structure, Amer. J. Phys. 50 (1991), 824-834. (1991) | MR 1126776
The holomorphic automorphism group of the complex disk, Aeq. Math. 47 (1994), 240-254. (1994) | MR 1268034 | Zbl 0799.20032
Categorical models and quasigroup homotopies, Elsevier Preprint, 1999. | MR 1988069 | Zbl 1050.20047