JOURNAL BROWSE
Search
Advanced SearchSearch Tips
SOME OPEN PROBLEMS IN THE THEORY OF INFINITE DIMENSIONAL ALGEBRAS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
SOME OPEN PROBLEMS IN THE THEORY OF INFINITE DIMENSIONAL ALGEBRAS
Zelmanov, Efim;
  PDF(new window)
 Abstract
We will discuss some very old and some new open problems concerning infinite dimensional algebras. All these problems have been inspired by combinatorial group theory.
 Keywords
infinite dimensional algebra;group;Gelfand-Kirillov dimension;
 Language
English
 Cited by
1.
Periodic Algebras Generated by Groups, Algebra Colloquium, 2015, 22, 04, 541  crossref(new windwow)
2.
GROWTH OF MODULES OVER GENERIC GOLOD-SHAFAREVICH-ALGEBRAS, Asian-European Journal of Mathematics, 2009, 02, 02, 279  crossref(new windwow)
3.
On the nilpotency degree of the algebra with identity, Journal of Algebra, 2012, 371, 350  crossref(new windwow)
4.
Golod–Shafarevich algebras, free subalgebras and Noetherian images, Journal of Algebra, 2013, 381, 116  crossref(new windwow)
5.
Representation Spaces of the Jordan Plane, Communications in Algebra, 2014, 42, 8, 3507  crossref(new windwow)
6.
GK–DIMENSION OF ALGEBRAS WITH MANY GENERIC RELATIONS*, Glasgow Mathematical Journal, 2009, 51, 02, 253  crossref(new windwow)
7.
Optimal 5-step nilpotent quadratic algebras, Journal of Algebra, 2014, 412, 1  crossref(new windwow)
8.
Nil algebras with restricted growth, Proceedings of the Edinburgh Mathematical Society, 2012, 55, 02, 461  crossref(new windwow)
9.
The Kurosh problem for Jordan nil systems over arbitrary rings of scalars, Journal of Algebra, 2015, 444, 313  crossref(new windwow)
10.
On normal subgroups of division rings which are radical over a proper division subring, Studia Scientiarum Mathematicarum Hungarica, 2014, 51, 2, 231  crossref(new windwow)
11.
Asymptotically optimal k-step nilpotency of quadratic algebras and the Fibonacci numbers, Combinatorica, 2016  crossref(new windwow)
12.
Growth, entropy and commutativity of algebras satisfying prescribed relations, Selecta Mathematica, 2014, 20, 4, 1197  crossref(new windwow)
13.
Three dimensional Sklyanin algebras and Gröbner bases, Journal of Algebra, 2017, 470, 379  crossref(new windwow)
14.
On groups with cubic polynomial conditions, Journal of Algebra, 2015, 437, 344  crossref(new windwow)
15.
Minimal Spectrum and the Radical of Chinese Algebras, Algebras and Representation Theory, 2013, 16, 4, 905  crossref(new windwow)
16.
COMPUTING NILPOTENT QUOTIENTS OF ASSOCIATIVE ALGEBRAS AND ALGEBRAS SATISFYING A POLYNOMIAL IDENTITY, International Journal of Algebra and Computation, 2011, 21, 08, 1339  crossref(new windwow)
17.
Finite dimensional semigroup quadratic algebras with the minimal number of relations, Monatshefte für Mathematik, 2012, 168, 2, 239  crossref(new windwow)
18.
Fractal nil graded Lie superalgebras, Journal of Algebra, 2016, 466, 229  crossref(new windwow)
19.
GOLOD–SHAFAREVICH GROUPS: A SURVEY, International Journal of Algebra and Computation, 2012, 22, 05, 1230001  crossref(new windwow)
20.
Jacobson radical non-nil algebras of Gel’fand-Kirillov dimension 2, Israel Journal of Mathematics, 2013, 194, 2, 597  crossref(new windwow)
 References
1.
S. V. Aljoshin, Finite automata and the Burnside problem for periodic groups, Mat. Zametki 11 (1972), 319-328

2.
S. A. Amitsur, Jacob Levitzki, 1904-1956, Israel J. Math. 19 (1974), 1-2

3.
D. J. Anick, Generic algebras and CW complexes, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), 247-321, Ann. of Math. Stud., 113, Princeton Univ. Press, Princeton, NJ, 1987

4.
J. Backelin, On the rates of growth of the homologies of Veronese subrings, Lecture Notes in Math., 1183, Springer, Berlin, 1986

5.
B. Barak, R. Impagliazzo, A. Shpilka, and A Wigderson, Private communication

6.
W. Burnside, On an unsettled question in the theory of discontinuous groups, Q. J. Pure Appl, Math. 33 (1902), 230-238

7.
W. Burnside, Theory of groups of finite order, Cambridge Univ. Press, 1911

8.
L. Carbone and H. Garland, Existence of lattices in Kac-Moody groups over finite fields, Commun. Contemp. Math. 5 (2003), no. 5,813-867 crossref(new window)

9.
V. A. Efremovic, The proximity geometry of Riemannian manifolds, Uspekhi Math. Nauk 8 (1953), 189

10.
M. Ershov, Golod-Shafarevich groups with property (T) and Kac-Moody groups, preprint

11.
I. M. Gelfand and A. A. Kirillov, Surles corps lies aux alqebres enveloppantes des alqebres de Lie, Inst. Hautes Etudes Sci. Publ. Math. No. 31 (1966), 5-19

12.
E. S. Golod, On nil-algebras and finitely approximable p-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273-276

13.
E. S. Golod and I. R. Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261-272

14.
R. I. Grigorchuk, On the Milnor problem of group growth, Dokl. Akad. Nauk SSSR 271 (1983), no. 1, 30-33

15.
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci. Publ. Math. No. 53 (1981), 53-73 crossref(new window)

16.
N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), no. 3, 385-388 crossref(new window)

17.
M. Hall, Solution of the Burnside problem for exponent six, Illinois J. Math. 2 (1958), 764-786

18.
S. Hoory, N. Linial, and A. Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439-561 crossref(new window)

19.
S. Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994), no. 1-2, 1-308 crossref(new window)

20.
N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math. (2) 46 (1945), 695-707 crossref(new window)

21.
I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575-580 crossref(new window)

22.
D. Kazhdan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Prilozen, 1 (1967), 71-74

23.
G. R. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition. Graduate Studies in Mathematics, 22, American Mathematical Society, Providence, RI, 2000

24.
A. G. Kurosh, Problems in ring theory which are related to the Burnside Problem for periodic groups, Izv. Akad. Nauk SSSR 5, no. 3 (1941), 233-240

25.
M. Lackenby, Large groups, property ($\tau$) and the homology growth of subgroups, preprint

26.
J. Levitzky, On a problem of A. Kurosch, Bull. Amer. Math. Soc. 52 (1946), 1033-1035 crossref(new window)

27.
T. H. Lenagan and A. Smoktunowicz, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension, preprint

28.
A. Lubotzky, Group presentation, p-adic analytic groups and lattices in $SL_2$(C), Ann. of Math. (2) 118 (1983), no. 1, 115-130 crossref(new window)

29.
A. Lubotzky, Eigenvalues of the Laplacian, the first Betti number and the congruence sub-group problem, Ann. of Math. (2) 144 (1996), no. 2,441-452 crossref(new window)

30.
A. Lubotzky , Discrete groups, expanding graphs and invariant measures, Progress in Mathematics, 125, Birkhauser Verlag, Basel, 1994

31.
A. Lubotzky and E. Zelmanov, Dimension expanders, to appear in J. Algebra

32.
Yu, I. Manin, Quantum groups and noncommutative geometry, Universite de Montreal, Centre de Recherches Mathematiques, Montreal, QC, 1988

33.
G. A. Margulis, Explicit constructions of expanders, Problemy Peredaci Informacii 9 (1973), no. 4, 71-80

34.
J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1-7

35.
A. Yu. Ol'shansky, The geometry of defining relations in groups, Nauka, Moscow, 1989

36.
A. Yu. Ol'shansky and M. V. Sapir, Non-amenable finitely presented torsion-by-cyclic groups, Publ. Math. Inst. Hautes Etudes Sci. No. 96 (2002), 43-169

37.
V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra 302 (2006), no. 2,881-886 crossref(new window)

38.
M. S. Pinsker, On the complexity of a concentrator, In 7th International Telegraffic Conference, 4 (1973), 1-318

39.
A. Polishchuk and L. Positelski, Quadratic algebras, University Lecture Series, 37, American Mathematical Society, Providence, RI, 2005

40.
B. Remy, Groupes de Kac-Moody deployes et presque deployes, Asterisque No. 277 (2002), 348 pp

41.
P. Roquette, On class field towers, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 231-249

42.
I. N. Sanoy, On a certain system of relations in periodic groups with period a power of a prime number, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 477-502

43.
I. Schur, Uber Gruppen periodischer linearer substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1911), 619-627

44.
A. S. Schwarze, A volume invariant of coverings, Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), 32-34

45.
I. R. Shafarevich, Extensions with prescribed ramification points, Inst. Hautes Etudes Sci. Publ. Math. No. 18 (1963), 71-95

46.
A.I. Shirshov, On rings with identity relations, Mat. Sb. N.S. 43(85) (1957), 277-283

47.
M. K. Smith, Universal enveloping algebras with subexponential but not polynomially bounded growth, Proc. Amer. Math. Soc. 60 (1976), 22-24 crossref(new window)

48.
V. I. Sushchansky, Periodic p-groups of permutations and the unrestricted Burnside problem, Dokl. Akad. Nauk SSSR 247 (1979), no. 3, 557-561

49.
I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, to appear

50.
E. B. Vinberg, On the theorem concerning the infinite-dimensionality of an associative algebra, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 209-214

51.
M. Vaughan-Lee, The restricted Burnside problem, Second edition. London Mathematical Society Monographs. New Series, 8, The Clarendon Press, Oxford University Press, New York, 1993

52.
T. Voden, Subalgebras of Golod-Shafarevich algebras, preprint

53.
A. Wigderson, personal communication

54.
J. Wilson, Finite presentations of pro-p groups and discrete groups, Invent. Math. 105 (1991), no. 1, 177-183 crossref(new window)

55.
E. Zelmanov, Solution of the restricted Burnside problem for groups of odd exponent, Math. USSR-Izv. 36 (1991), no. 1,41-60 crossref(new window)

56.
E. Zelmanov , Solution of the restricted Burnside problem for 2-groups, Math. USSR-Sb. 72 (1992), no. 2, 543-565 crossref(new window)

57.
E. Zelmanov, Lie ring methods in the theory of nilpotent groups, Groups '93 Galway 1St. Andrews, Vol. 2, 567-585, London Math. Soc. Lecture Note Ser., 212, Cambridge Univ. Press, Cambridge, 1995

58.
E. Zelmanov, On groups satisfying the Golod-Shafarevich condition, New horizons in pro-p groups, 223-232, Progr. Math., 184, Birkhauser Boston, Boston, MA, 2000