The moduli space of \(1|1\)-dimensional complex associative algebras. (English) Zbl 1285.16025
From the introduction: In this paper, we will give a complete description of the moduli space of \(1|1\)-dimensional algebras, including a computation of a miniversal deformation of each of these algebras. From the miniversal deformations, a decomposition of the moduli space into strata is obtained, with the only connections between strata given by jump deformations. In the \(1|1\)-dimensional case, the description is simple, because each of the strata consists of a single point, so the only interesting information is given by the jump deformations.
The main result of this paper is the complete description of the cohomology for all \(1|1\)-dimensional associative algebras. It turns out that the calculation of cohomology even for low dimensional associative algebras is a non-trivial problem. To construct extensions of associative algebras to \(A_\infty\) algebras, it is necessary to have a complete description of the cohomology in all degrees, not just \(H^2\) and \(H^3\), which are needed for the deformation theory of these algebras as associative algebras. What we compute in this paper is the first step in constructing \(1|1\)-dimensional \(A_\infty\) algebras. These results may be of interest on their own, especially as an indication of the difficulty which occurs in computing the deformation theory of associative algebras, even in low dimension.
The versal deformation of an associative algebra depends only on the second and third Hochschild cohomology groups. However, we give a complete calculation of the cohomology for each of the algebras. What makes the study of associative algebras of low dimension much more complicated than the corresponding study of low dimensional Lie algebras is that while for a Lie algebra, the Hochschild cohomology \(H^n\) vanishes for \(n\) larger than the dimension of the vector space, in general, for an associative algebra \(H^n\) does not vanish. Thus we had to develop arguments on a case-by-case basis for each of the six distinct algebras. In particular, one of these algebras has an unusual pattern for the cohomology, which made its computation rather nontrivial.
The main result of this paper is the complete description of the cohomology for all \(1|1\)-dimensional associative algebras. It turns out that the calculation of cohomology even for low dimensional associative algebras is a non-trivial problem. To construct extensions of associative algebras to \(A_\infty\) algebras, it is necessary to have a complete description of the cohomology in all degrees, not just \(H^2\) and \(H^3\), which are needed for the deformation theory of these algebras as associative algebras. What we compute in this paper is the first step in constructing \(1|1\)-dimensional \(A_\infty\) algebras. These results may be of interest on their own, especially as an indication of the difficulty which occurs in computing the deformation theory of associative algebras, even in low dimension.
The versal deformation of an associative algebra depends only on the second and third Hochschild cohomology groups. However, we give a complete calculation of the cohomology for each of the algebras. What makes the study of associative algebras of low dimension much more complicated than the corresponding study of low dimensional Lie algebras is that while for a Lie algebra, the Hochschild cohomology \(H^n\) vanishes for \(n\) larger than the dimension of the vector space, in general, for an associative algebra \(H^n\) does not vanish. Thus we had to develop arguments on a case-by-case basis for each of the six distinct algebras. In particular, one of these algebras has an unusual pattern for the cohomology, which made its computation rather nontrivial.
MSC:
| 16S80 | Deformations of associative rings |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16W55 | “Super” (or “skew”) structure |
References:
| [1] | DOI: 10.4310/HHA.2005.v7.n2.a3 · Zbl 1152.17306 · doi:10.4310/HHA.2005.v7.n2.a3 |
| [2] | DOI: 10.1080/00927870701665339 · Zbl 1138.14009 · doi:10.1080/00927870701665339 |
| [3] | DOI: 10.1006/jfan.1998.3349 · Zbl 0944.17015 · doi:10.1006/jfan.1998.3349 |
| [4] | DOI: 10.1016/S0021-8693(02)00067-4 · Zbl 1038.17012 · doi:10.1016/S0021-8693(02)00067-4 |
| [5] | DOI: 10.1142/S0219199705001702 · Zbl 1075.14007 · doi:10.1142/S0219199705001702 |
| [6] | DOI: 10.1142/S0219199707002344 · Zbl 1127.14014 · doi:10.1142/S0219199707002344 |
| [7] | DOI: 10.1007/s10773-007-9481-4 · Zbl 1142.17003 · doi:10.1007/s10773-007-9481-4 |
| [8] | DOI: 10.1080/00927870902747845 · Zbl 1183.14019 · doi:10.1080/00927870902747845 |
| [9] | DOI: 10.2307/1970343 · Zbl 0131.27302 · doi:10.2307/1970343 |
| [10] | DOI: 10.2307/1970484 · Zbl 0123.03101 · doi:10.2307/1970484 |
| [11] | DOI: 10.2307/1970528 · Zbl 0147.28903 · doi:10.2307/1970528 |
| [12] | DOI: 10.2307/1970553 · Zbl 0182.05902 · doi:10.2307/1970553 |
| [13] | DOI: 10.2307/1970900 · Zbl 0281.16016 · doi:10.2307/1970900 |
| [14] | DOI: 10.2307/1969145 · Zbl 0063.02029 · doi:10.2307/1969145 |
| [15] | C. Otto and M. Penkava, Algebra and Its Applications (American Mathematical Society, 2007) pp. 255–269. |
| [16] | DOI: 10.1016/0022-4049(93)90096-C · Zbl 0786.57017 · doi:10.1016/0022-4049(93)90096-C |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
