Classification and derivations of low-dimensional complex dialgebras
The thesis is mainly comprised of two parts. In the first part we consider the classification problem of low-dimensional associative, diassociative and dendriform algebras. Since so far there are no research results dealing with representing diassociative and dendriform algebras in form of precis...
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| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2014
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/70472/1/FS%202014%2047%20-%20IR.pdf |
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| Summary: | The thesis is mainly comprised of two parts. In the first part we consider the
classification problem of low-dimensional associative, diassociative and dendriform
algebras. Since so far there are no research results dealing with representing diassociative
and dendriform algebras in form of precise tables under some basis, it is
desirable to have such lists up to isomorphisms. There is no standard approach to
the classification problem of algebras. One of the approaches which can be applied
is to fix a basis and represent the algebras in terms of structure constants. Due to
the identities we have constraints for the structure constants in polynomial form.
Solving the system of polynomials we get a redundant list of all the algebras from
given class. Then we erase isomorphic copies from the list. It is slightly tedious
to perform this procedure by hand. For this case we construct and use several
computer programs. They are applied to verify the isomorphism between found
algebras, to find automorphism groups and verify the algebra identities.
In conclusion, we give complete lists of isomorphism classes for diassociative and
dendriform algebras in low dimensions. We found for diassociative algebras four
isomorphism classes (one parametric family and another three are single class)
in dimension two, 17 isomorphism classes (one parametric family and others are
single classes) in dimension three and for nilpotent diassociative algebras we obtain
16 isomorphism classes (all of them are parametric family) in dimension four.
In dendriform algebras case there are twelve isomorphism classes (one parametric
family and another eleven are single classes) in dimension two.
The second part of the thesis is devoted to the computation of derivations of
low-dimensional associative, diassociative and dendriform algebras. We give the
derivations the above mentioned classes of algebras in dimensions two and three. |
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