American Institute of Mathematical Sciences

Advanced
  2017, 24: 78-86. doi: 10.3934/era.2017.24.009

On matrix wreath products of algebras

Adel Alahmadi 1, , Hamed Alsulami 1, , S.K. Jain 2, and  Efim Zelmanov 3,

1. 

Department of Mathematics, King Abdulaziz University, Jeddah, SA
2. 

Department of Mathematics, Ohio University, Athens, USA
3. 

Department of Mathematics, University of California, San Diego, USA

Received  April 23, 2017 Published  August 2017

Fund Project: The fourth author gratefully acknowledges the support from the NSF. The authors are grateful to the referees for numerous valuable comments

We introduce a new construction of matrix wreath products of algebras that is similar to the construction of wreath products of groups introduced by L. Kaloujnine and M. Krasner [17]. We then illustrate its usefulness by proving embedding theorems into finitely generated algebras and constructing nil algebras with prescribed Gelfand-Kirillov dimension.

Keywords: growth function, associative algebra, wreath product.
Mathematics Subject Classification: 16S99(Primary), 16P90(Secondary).
Citation: Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009
References:
[1]

A. Alahmadi and H. Alsulami, Wreath products by a Leavitt path algebra and affinizations, Internat. J. Algebra Comput., 24 (2014), 707-714.  doi: 10.1142/S0218196714500295.  Google Scholar

[2]

A. S. Amitsur, Algebras over infinite fields, Proc. Amer. Math. Soc., 7 (1956), 35-48.  doi: 10.1090/S0002-9939-1956-0075933-2.  Google Scholar

[3]

L. Bartholdi, Self-similar Lie algebras, J. Eur. Math. Soc. (JEMS), 17 (2015), 3113-3151.  doi: 10.4171/JEMS/581.  Google Scholar

[4]

L. Bartholdi and A. Erschler, Imbeddings into groups of intermediate growth, Groups Geom. Dyn., 8 (2014), 605-620.  doi: 10.4171/GGD/241.  Google Scholar

[5]

L. Bartholdi and A. Smoktunowicz, Images of Golod-Shafarevich algebras with small growth, Q. J. Math., 65 (2014), 421-438.  doi: 10.1093/qmath/hat005.  Google Scholar

[6]

J. P. Bell, Examples in finite Gel$\prime$ fand-Kirillov dimension, J. Algebra, 263 (2003), 159-175.  doi: 10.1016/S0021-8693(03)00021-8.  Google Scholar

[7]

J. P. Bell and L. W. Small, A question of Kaplansky, J. Algebra, 258 (2002), 386-388.  doi: 10.1016/S0021-8693(02)00513-6.  Google Scholar

[8]

J. P. Bell, L. W. Small and A. Smoktunowicz, Primitive algebraic algebras of polynomially bounded growth, in New Trends in Noncommutative Algebra, Contemp. Math., 562, Amer. Math. Soc., Providence, RI, 2012, 41–52. doi: 10.1090/conm/562/11129.  Google Scholar

[9]

K. I. Beĭ dar, Radicals of finitely generated algebras, Uspekhi Mat. Nauk, 36 (1981), 203-204.   Google Scholar

[10]

W. Borho and H. Kraft, über die Gelfand-Kirillov-Dimension, Math. Ann., 220 (1976), 1-24.  doi: 10.1007/BF01354525.  Google Scholar

[11]

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

[12]

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

[13]

B. Greenfeld, Prime and primitive algebras with prescribed growth types, Israel J. Math., 220 (2017), 161-174.  doi: 10.1007/s11856-017-1513-z.  Google Scholar

[14]

R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 939-985.   Google Scholar

[15]

G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc., 24 (1949), 247-254.  doi: 10.1112/jlms/s1-24.4.247.  Google Scholar

[16]

N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, Vol. 37, Revised edition, American Mathematical Society, Providence, R. I., 1964.  Google Scholar

[17]

L. Kaloujnine and M. Krasner, Le produit complet des groupes de permutations et le probléme d'extension des groupes, C. R. Acad. Sci. Paris, 227 (1948), 806-808.   Google Scholar

[18]

I. Kaplansky, ''Problems in the theory of rings'' revisited, Amer. Math. Monthly, 77 (1970), 445-454.  doi: 10.2307/2317376.  Google Scholar

[19]

T. H. Lenagan and A. Smoktunowicz, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension, J. Amer. Math. Soc., 20 (2007), 989-1001.  doi: 10.1090/S0894-0347-07-00565-6.  Google Scholar

[20]

T. H. Lenagan, A. Smoktunowicz and A. A. Young, Nil algebras with restricted growth, Proc. Edinb. Math. Soc.(2), 55 (2012), 461-475.  doi: 10.1017/S0013091510001100.  Google Scholar

[21]

A. I. Mal$\prime$ cev, On a representation of nonassociative rings, Uspehi Matem. Nauk (N.S.), 7 (1952), 181-185.   Google Scholar

[22]

V. T. Markov, Matrix algebras with two generators and the embedding of PI-algebras, Uspekhi Mat. Nauk, 47 (1992), 199-200.   Google Scholar

[23]

A. Yu. Olshanskii and D. V. Osin, A quasi-isometric embedding theorem for groups, Duke Math. J., 162 (2013), 1621-1648.  doi: 10.1215/00127094-2266251.  Google Scholar

[24]

V. M. Petrogradsky, Yu. P. Razmyslov and E. O. Shishkin, Wreath products and Kaluzhnin-Krasner embedding for Lie algebras, Proc. Amer. Math. Soc., 135 (2007), 625-636.  doi: 10.1090/S0002-9939-06-08502-9.  Google Scholar

[25]

R. E. Phillips, Embedding methods for periodic groups, Proc. London Math. Soc.(3), 35 (1977), 238-256.  doi: 10.1112/plms/s3-35.2.238.  Google Scholar

[26]

A. I. Siř sov, On free Lie rings, Mat. Sb. N.S., 45(87) (1958), 113-122.   Google Scholar

[27]

A. L. Smel'kin, Wreath products of Lie algebras, and their application in group theory, Trudy Moskov. Mat. Obšč., 29 (1973), 247-260.   Google Scholar

[28]

M. K. Smith, Universal enveloping algebras with subexponential but not polynomially bounded growth, Proc. Amer. Math. Soc., 60 (1976), 22-24 (1977).  doi: 10.1090/S0002-9939-1976-0419534-5.  Google Scholar

[29]

A. Smoktunowicz and L. Bartholdi, Jacobson radical non-nil algebras of Gel'fand-Kirillov dimension 2, Israel J. Math., 194 (2013), 597-608.  doi: 10.1007/s11856-012-0073-5.  Google Scholar

[30]

J. S. Wilson, Embedding theorems for residually finite groups, Math. Z., 174 (1980), 149-157.  doi: 10.1007/BF01293535.  Google Scholar

show all references

References:
[1]

A. Alahmadi and H. Alsulami, Wreath products by a Leavitt path algebra and affinizations, Internat. J. Algebra Comput., 24 (2014), 707-714.  doi: 10.1142/S0218196714500295.  Google Scholar

[2]

A. S. Amitsur, Algebras over infinite fields, Proc. Amer. Math. Soc., 7 (1956), 35-48.  doi: 10.1090/S0002-9939-1956-0075933-2.  Google Scholar

[3]

L. Bartholdi, Self-similar Lie algebras, J. Eur. Math. Soc. (JEMS), 17 (2015), 3113-3151.  doi: 10.4171/JEMS/581.  Google Scholar

[4]

L. Bartholdi and A. Erschler, Imbeddings into groups of intermediate growth, Groups Geom. Dyn., 8 (2014), 605-620.  doi: 10.4171/GGD/241.  Google Scholar

[5]

L. Bartholdi and A. Smoktunowicz, Images of Golod-Shafarevich algebras with small growth, Q. J. Math., 65 (2014), 421-438.  doi: 10.1093/qmath/hat005.  Google Scholar

[6]

J. P. Bell, Examples in finite Gel$\prime$ fand-Kirillov dimension, J. Algebra, 263 (2003), 159-175.  doi: 10.1016/S0021-8693(03)00021-8.  Google Scholar

[7]

J. P. Bell and L. W. Small, A question of Kaplansky, J. Algebra, 258 (2002), 386-388.  doi: 10.1016/S0021-8693(02)00513-6.  Google Scholar

[8]

J. P. Bell, L. W. Small and A. Smoktunowicz, Primitive algebraic algebras of polynomially bounded growth, in New Trends in Noncommutative Algebra, Contemp. Math., 562, Amer. Math. Soc., Providence, RI, 2012, 41–52. doi: 10.1090/conm/562/11129.  Google Scholar

[9]

K. I. Beĭ dar, Radicals of finitely generated algebras, Uspekhi Mat. Nauk, 36 (1981), 203-204.   Google Scholar

[10]

W. Borho and H. Kraft, über die Gelfand-Kirillov-Dimension, Math. Ann., 220 (1976), 1-24.  doi: 10.1007/BF01354525.  Google Scholar

[11]

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

[12]

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

[13]

B. Greenfeld, Prime and primitive algebras with prescribed growth types, Israel J. Math., 220 (2017), 161-174.  doi: 10.1007/s11856-017-1513-z.  Google Scholar

[14]

R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat., 48 (1984), 939-985.   Google Scholar

[15]

G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc., 24 (1949), 247-254.  doi: 10.1112/jlms/s1-24.4.247.  Google Scholar

[16]

N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, Vol. 37, Revised edition, American Mathematical Society, Providence, R. I., 1964.  Google Scholar

[17]

L. Kaloujnine and M. Krasner, Le produit complet des groupes de permutations et le probléme d'extension des groupes, C. R. Acad. Sci. Paris, 227 (1948), 806-808.   Google Scholar

[18]

I. Kaplansky, ''Problems in the theory of rings'' revisited, Amer. Math. Monthly, 77 (1970), 445-454.  doi: 10.2307/2317376.  Google Scholar

[19]

T. H. Lenagan and A. Smoktunowicz, An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension, J. Amer. Math. Soc., 20 (2007), 989-1001.  doi: 10.1090/S0894-0347-07-00565-6.  Google Scholar

[20]

T. H. Lenagan, A. Smoktunowicz and A. A. Young, Nil algebras with restricted growth, Proc. Edinb. Math. Soc.(2), 55 (2012), 461-475.  doi: 10.1017/S0013091510001100.  Google Scholar

[21]

A. I. Mal$\prime$ cev, On a representation of nonassociative rings, Uspehi Matem. Nauk (N.S.), 7 (1952), 181-185.   Google Scholar

[22]

V. T. Markov, Matrix algebras with two generators and the embedding of PI-algebras, Uspekhi Mat. Nauk, 47 (1992), 199-200.   Google Scholar

[23]

A. Yu. Olshanskii and D. V. Osin, A quasi-isometric embedding theorem for groups, Duke Math. J., 162 (2013), 1621-1648.  doi: 10.1215/00127094-2266251.  Google Scholar

[24]

V. M. Petrogradsky, Yu. P. Razmyslov and E. O. Shishkin, Wreath products and Kaluzhnin-Krasner embedding for Lie algebras, Proc. Amer. Math. Soc., 135 (2007), 625-636.  doi: 10.1090/S0002-9939-06-08502-9.  Google Scholar

[25]

R. E. Phillips, Embedding methods for periodic groups, Proc. London Math. Soc.(3), 35 (1977), 238-256.  doi: 10.1112/plms/s3-35.2.238.  Google Scholar

[26]

A. I. Siř sov, On free Lie rings, Mat. Sb. N.S., 45(87) (1958), 113-122.   Google Scholar

[27]

A. L. Smel'kin, Wreath products of Lie algebras, and their application in group theory, Trudy Moskov. Mat. Obšč., 29 (1973), 247-260.   Google Scholar

[28]

M. K. Smith, Universal enveloping algebras with subexponential but not polynomially bounded growth, Proc. Amer. Math. Soc., 60 (1976), 22-24 (1977).  doi: 10.1090/S0002-9939-1976-0419534-5.  Google Scholar

[29]

A. Smoktunowicz and L. Bartholdi, Jacobson radical non-nil algebras of Gel'fand-Kirillov dimension 2, Israel J. Math., 194 (2013), 597-608.  doi: 10.1007/s11856-012-0073-5.  Google Scholar

[30]

J. S. Wilson, Embedding theorems for residually finite groups, Math. Z., 174 (1980), 149-157.  doi: 10.1007/BF01293535.  Google Scholar

[1]

Joseph Bayara, André Conseibo, Artibano Micali, Moussa Ouattara. Derivations in power-associative algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1359-1370. doi: 10.3934/dcdss.2011.4.1359
[2]

Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-15. doi: 10.3934/dcdss.2020066
[3]

Hari Bercovici, Viorel Niţică. A Banach algebra version of the Livsic theorem. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 523-534. doi: 10.3934/dcds.1998.4.523
[4]

Jianhong Wu, Ruyuan Zhang. A simple delayed neural network with large capacity for associative memory. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 851-863. doi: 10.3934/dcdsb.2004.4.851
[5]

K. L. Mak, J. G. Peng, Z. B. Xu, K. F. C. Yiu. A novel neural network for associative memory via dynamical systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 573-590. doi: 10.3934/dcdsb.2006.6.573
[6]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420
[7]

Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.

[8]

Viktor Levandovskyy, Gerhard Pfister, Valery G. Romanovski. Evaluating cyclicity of cubic systems with algorithms of computational algebra. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2023-2035. doi: 10.3934/cpaa.2012.11.2023
[9]

Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399
[10]

Stefano Luzzatto, Marks Ruziboev. Young towers for product systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1465-1491. doi: 10.3934/dcds.2016.36.1465
[11]

Nir Avni, Benjamin Weiss. Generating product systems. Journal of Modern Dynamics, 2010, 4 (2) : 257-270. doi: 10.3934/jmd.2010.4.257
[12]

A. V. Grishin. On non-Spechtianness of the variety of associative rings that satisfy the identity $x^{32} = 0$. Electronic Research Announcements, 2000, 6: 50-51.

[13]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Convolutional codes with a matrix-algebra word-ambient. Advances in Mathematics of Communications, 2016, 10 (1) : 29-43. doi: 10.3934/amc.2016.10.29
[14]

H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50
[15]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239
[16]

A. S. Dzhumadil'daev. Jordan elements and Left-Center of a Free Leibniz algebra. Electronic Research Announcements, 2011, 18: 31-49. doi: 10.3934/era.2011.18.31
[17]

Navin Keswani. Homotopy invariance of relative eta-invariants and $C^*$-algebra $K$-theory. Electronic Research Announcements, 1998, 4: 18-26.

[18]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
[19]

Boris Hasselblatt and Jorg Schmeling. Dimension product structure of hyperbolic sets. Electronic Research Announcements, 2004, 10: 88-96.

[20]

Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252

2018 Impact Factor: 0.263

Tools

Article outline

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences