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

On matrix wreath products of algebras

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.

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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. 

[15]

G. HigmanB. 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.

[16]

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

[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. 

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

V. M. PetrogradskyYu. 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.

[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.

[26]

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

[27]

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

[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.

[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.

[30]

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

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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. 

[15]

G. HigmanB. 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.

[16]

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

[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. 

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[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.

[24]

V. M. PetrogradskyYu. 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.

[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.

[26]

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

[27]

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

[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.

[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.

[30]

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

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