DOI QR코드

DOI QR Code

SOME OPEN PROBLEMS IN THE THEORY OF INFINITE DIMENSIONAL ALGEBRAS

  • Zelmanov, Efim (Department of Mathematics University of California at San Diego)
  • Published : 2007.09.30

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

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 https://doi.org/10.1142/S0219199703001117
  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 https://doi.org/10.1007/BF02698687
  16. N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Math. Z. 182 (1983), no. 3, 385-388 https://doi.org/10.1007/BF01179757
  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 https://doi.org/10.1090/S0273-0979-06-01126-8
  19. S. Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994), no. 1-2, 1-308 https://doi.org/10.1142/S0218196794000026
  20. N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math. (2) 46 (1945), 695-707 https://doi.org/10.2307/1969205
  21. I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575-580 https://doi.org/10.1090/S0002-9904-1948-09049-8
  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 https://doi.org/10.1090/S0002-9904-1946-08705-4
  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 https://doi.org/10.2307/2006956
  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 https://doi.org/10.2307/2118597
  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 https://doi.org/10.1016/j.jalgebra.2005.09.005
  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 https://doi.org/10.2307/2041103
  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 https://doi.org/10.1007/BF01232262
  55. E. Zelmanov, Solution of the restricted Burnside problem for groups of odd exponent, Math. USSR-Izv. 36 (1991), no. 1,41-60 https://doi.org/10.1070/IM1991v036n01ABEH001946
  56. E. Zelmanov , Solution of the restricted Burnside problem for 2-groups, Math. USSR-Sb. 72 (1992), no. 2, 543-565 https://doi.org/10.1070/SM1992v072n02ABEH001272
  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

Cited by

  1. GOLOD–SHAFAREVICH GROUPS: A SURVEY vol.22, pp.05, 2012, https://doi.org/10.1142/S0218196712300010
  2. Asymptotically optimal k-step nilpotency of quadratic algebras and the Fibonacci numbers vol.37, pp.3, 2017, https://doi.org/10.1007/s00493-016-3009-6
  3. The LNED and LFED conjectures for algebraic algebras vol.534, 2017, https://doi.org/10.1016/j.laa.2017.08.016
  4. Growth, entropy and commutativity of algebras satisfying prescribed relations vol.20, pp.4, 2014, https://doi.org/10.1007/s00029-014-0154-x
  5. Nil algebras with restricted growth vol.55, pp.02, 2012, https://doi.org/10.1017/S0013091510001100
  6. Three dimensional Sklyanin algebras and Gröbner bases vol.470, 2017, https://doi.org/10.1016/j.jalgebra.2016.08.023
  7. On normal subgroups of division rings which are radical over a proper division subring vol.51, pp.2, 2014, https://doi.org/10.1556/SScMath.51.2014.2.1265
  8. GROWTH OF MODULES OVER GENERIC GOLOD-SHAFAREVICH-ALGEBRAS vol.02, pp.02, 2009, https://doi.org/10.1142/S1793557109000224
  9. COMPUTING NILPOTENT QUOTIENTS OF ASSOCIATIVE ALGEBRAS AND ALGEBRAS SATISFYING A POLYNOMIAL IDENTITY vol.21, pp.08, 2011, https://doi.org/10.1142/S0218196711006649
  10. Jacobson radical non-nil algebras of Gel’fand-Kirillov dimension 2 vol.194, pp.2, 2013, https://doi.org/10.1007/s11856-012-0073-5
  11. Nil Lie p-algebras of slow growth vol.45, pp.7, 2017, https://doi.org/10.1080/00927872.2016.1233233
  12. Golod–Shafarevich algebras, free subalgebras and Noetherian images vol.381, 2013, https://doi.org/10.1016/j.jalgebra.2013.02.003
  13. Optimal 5-step nilpotent quadratic algebras vol.412, 2014, https://doi.org/10.1016/j.jalgebra.2014.04.015
  14. On groups with cubic polynomial conditions vol.437, 2015, https://doi.org/10.1016/j.jalgebra.2015.04.035
  15. The Kurosh problem for Jordan nil systems over arbitrary rings of scalars vol.444, 2015, https://doi.org/10.1016/j.jalgebra.2015.08.004
  16. Minimal Spectrum and the Radical of Chinese Algebras vol.16, pp.4, 2013, https://doi.org/10.1007/s10468-012-9339-1
  17. GK–DIMENSION OF ALGEBRAS WITH MANY GENERIC RELATIONS* vol.51, pp.02, 2009, https://doi.org/10.1017/S0017089508004667
  18. Fractal nil graded Lie superalgebras vol.466, 2016, https://doi.org/10.1016/j.jalgebra.2016.06.028
  19. On the nilpotency degree of the algebra with identity xn=0 vol.371, 2012, https://doi.org/10.1016/j.jalgebra.2012.08.007
  20. Representation Spaces of the Jordan Plane vol.42, pp.8, 2014, https://doi.org/10.1080/00927872.2013.788184
  21. Finite dimensional semigroup quadratic algebras with the minimal number of relations vol.168, pp.2, 2012, https://doi.org/10.1007/s00605-011-0339-8
  22. Periodic Algebras Generated by Groups vol.22, pp.04, 2015, https://doi.org/10.1142/S1005386715000462
  23. On graded characterizations of finite dimensionality for algebraic algebras 2017, https://doi.org/10.1007/s00013-017-1090-8
  24. Golod–Shafarevich-Type Theorems and Potential Algebras pp.1687-0247, 2018, https://doi.org/10.1093/imrn/rnx315
  25. On Weakly Locally Finite Division Rings pp.2315-4144, 2018, https://doi.org/10.1007/s40306-018-0292-x