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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 |
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 [
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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