Application of Annihilator Extension's Method to Classify Zinbiel Algebras

یخصص المشروع لتطبیق تقنیة التمدیدات إلى فئة من الجبریات تسمى بجبریات زنبیل. دوافع البحث ھي عدم وجود نتائج للتصنیفات الأساسیة لھذه الفئة جبریات زنبیل من الجبر، خلافا لفئات أخرى من الجبر محدودة الأبعاد ترتبط ارتباطا وثیقا بھذه الفئة. بدایةً سنعمل على تطویر خلفیة نظریة لطریقة التمدید ثم تطبیقھا لتصنیف جبریات زنبیل وصولاً إلى البعد الخامس. الفكرة الرئیسیة من وراء طریقة التمدید ھي بناء الجبریات في الأبعاد العالیة ھذه الطریقة تمكننا من استخدام برامج الكمبیوتر لبعض الحسابات. النھج المقترح في البحث ھو نھج جدید، یعطي قائمة كاملة من جبریات زنبیل في البعد الخامس.

The project is devoted to application of extensions technique to a class of algebras Zinbiel called algebras. The research motivated by lack of essential classification results for this class of algebras unlike for classes of finitedimensional algebras closely related to this class. First, we develop a theoretical background of the extension method then apply it to classify Zinbiel algebras in dimension up to 5. Classification methods applied earlier were based on study of the behaviour of structure constants under base change and they were applicable in low-dimensional cases only. Main idea behind of the extension method is construction of algebras in higher dimension having lists of isomorphism classes in low dimensions which are available due to structure constants technique mentioned above. The method enables us to use computer programs for some computations. In all the cases we propose 2-cocycles and respective annihilator extensions. The base change action is interpreted as an action of automorphism group of small algebras on cocycles. The approach proposed is new, gives complete list of Zinbiel algebras in dimension 5. Note that all the existing classification results of Zinbiel algebras are supposed to be over the field of complex numbers whereas the approach proposed in the project deals with algebras over any algebraic closed field of characteristic not two.