Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Infinite homogeneous algebras are anticommutative

Author(s): Dragomir Z. Dokovic; Lowell G. Sweet
Journal: Proc. Amer. Math. Soc. 127 (1999), 3169-3174.
MSC (1991): Primary 17D99; Secondary 17A36, 15A69
Posted: May 13, 1999
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: A (non-associative) algebra $A$, over a field $k$, is called homogeneous if its automorphism group permutes transitively the one dimensional subspaces of $A$. Suppose $A$ is a nontrivial finite dimensional homogeneous algebra over an infinite field. Then we prove that $x^{2}=0$ for all $x$ in $A$, and so $xy=-yx$ for all $x,y\in A$.

References:

1.
V. A. Artamonov, On algebras without proper subalgebras, Math. USSR Sbornik 33 (1977), 375-401.

2.
J. Boen, O. Rothaus and J. Thompson, Further results on p-automorphic p-groups, Pacific J. Math. 12 (1962), 817-821. MR 27:25536

3.
D. \v{Z}. Djokovi\'{c}, Real homogeneous algebras, Proc. Amer. Math. Soc. 41 (1973), 457-462. MR 48:11227

4.
F. Gross, Finite automorphic algebras over $\mathrm{GF}(2)$, Proc. Amer. Math. Soc. 4 (1971), 10-14. MR 44:4063

5.
D. N. Ivanov, On homogeneous algebras over $\mathrm{GF}(2)$, Vestnik Moskov. Univ. Matematika 37 (1982), 69-72. MR 83k:17003

6.
A. I. Kostrikin, On homogeneous algebras, Izvestiya Akad. Nauk. USSR 29 (1965), 471-484. MR 31:219

7.
J. A. MacDougall and L. G. Sweet, Three dimensional homogeneous algebras, Pacific J. Math. 74 (1978), 153-162. MR 57:6121

8.
L. G. Sweet and J. A. MacDougall, Four dimensional homogeneous algebras, Pacific J. Math. 129(2) (1987), 375-383. MR 89c:17003

9.
L. G. Sweet, On homogeneous algebras, Pacific J. Math. 59 (1975), 585-594. MR 52:8202

10.
L. G. Sweet, On the triviality of homogeneous algebras over an algebraically closed field, Proc. Amer. Math. Soc. 48 (1975), 321-324. MR 51:637

11.
E. E. Shult, On the triviality of finite automorphic algebras, Illinois J. Math. 13 (1969), 654-659. MR 40:1442

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 17D99, 17A36, 15A69

Retrieve articles in all Journals with MSC (1991): 17D99, 17A36, 15A69

Additional Information:

Dragomir Z. Dokovic
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: dragomir@herod.uwaterloo.ca

Lowell G. Sweet
Affiliation: Department of Mathematics, University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada C1A 4P3
Email: sweet@upei.ca

DOI: 10.1090/S0002-9939-99-04910-2
PII: S 0002-9939(99)04910-2
Keywords: Non-associative algebras, automorphism group, hypersurface
Received by editor(s): January 7, 1998
Received by editor(s) in revised form: February 6, 1998
Posted: May 13, 1999
Additional Notes: This work was supported in part by the NSERC Grant A-5285.
Communicated by: Lance W. Small
Copyright of article: Copyright 1999, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google