Details
| Original language | English |
|---|---|
| Pages (from-to) | 791-802 |
| Number of pages | 12 |
| Journal | Journal of algebra |
| Volume | 301 |
| Issue number | 2 |
| Publication status | Published - 15 Jul 2006 |
| Externally published | Yes |
Abstract
The representation dimension of an Artin algebra was defined by M. Auslander in 1970. The precise value is not known in general, and is very hard to compute even for small examples. For group algebras, it is known in the case of cyclic Sylow subgroups. For some group algebras (in characteristic 2) of rank at least 3 the precise value of the representation dimension follows from recent work of R. Rouquier. There is a gap for group algebras of rank 2. In this paper we show that for all n {greater than or slanted equal to} 0 and any field k the commutative algebras k [ x, y ] / ( x2, y2 + n ) have representation dimension 3. For the proof, we give an explicit inductive construction of a suitable generator-cogenerator. As a consequence, we obtain that the group algebras in characteristic 2 of the groups C2 × C2m have representation dimension 3. Note that for m {greater than or slanted equal to} 3 these group algebras have wild representation type.
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of algebra, Vol. 301, No. 2, 15.07.2006, p. 791-802.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The representation dimension of k [ x, y ] / ( x2, yn )
AU - Holm, Thorsten
AU - Hu, Wei
N1 - Funding Information: ✩ The authors gratefully acknowledge financial support by the AsiaLink network Algebras and Representations in China and Europe, contract number ASI/B7-301/98/679-11. The second author is also supported by the Doctor Program Foundation (No. 20040027002), Ministry of Education of China. * Corresponding author. E-mail addresses: tholm@maths.leeds.ac.uk (T. Holm), hwxbest@163.com (W. Hu).
PY - 2006/7/15
Y1 - 2006/7/15
N2 - The representation dimension of an Artin algebra was defined by M. Auslander in 1970. The precise value is not known in general, and is very hard to compute even for small examples. For group algebras, it is known in the case of cyclic Sylow subgroups. For some group algebras (in characteristic 2) of rank at least 3 the precise value of the representation dimension follows from recent work of R. Rouquier. There is a gap for group algebras of rank 2. In this paper we show that for all n {greater than or slanted equal to} 0 and any field k the commutative algebras k [ x, y ] / ( x2, y2 + n ) have representation dimension 3. For the proof, we give an explicit inductive construction of a suitable generator-cogenerator. As a consequence, we obtain that the group algebras in characteristic 2 of the groups C2 × C2m have representation dimension 3. Note that for m {greater than or slanted equal to} 3 these group algebras have wild representation type.
AB - The representation dimension of an Artin algebra was defined by M. Auslander in 1970. The precise value is not known in general, and is very hard to compute even for small examples. For group algebras, it is known in the case of cyclic Sylow subgroups. For some group algebras (in characteristic 2) of rank at least 3 the precise value of the representation dimension follows from recent work of R. Rouquier. There is a gap for group algebras of rank 2. In this paper we show that for all n {greater than or slanted equal to} 0 and any field k the commutative algebras k [ x, y ] / ( x2, y2 + n ) have representation dimension 3. For the proof, we give an explicit inductive construction of a suitable generator-cogenerator. As a consequence, we obtain that the group algebras in characteristic 2 of the groups C2 × C2m have representation dimension 3. Note that for m {greater than or slanted equal to} 3 these group algebras have wild representation type.
UR - http://www.scopus.com/inward/record.url?scp=33747163262&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2005.11.037
DO - 10.1016/j.jalgebra.2005.11.037
M3 - Article
AN - SCOPUS:33747163262
VL - 301
SP - 791
EP - 802
JO - Journal of algebra
JF - Journal of algebra
SN - 0021-8693
IS - 2
ER -