American Institute of Mathematical Sciences

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  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

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