Mathematical Researches

fa توسیع مرکزی جامع برای سوپرجبر لی حاصل از ضرب تانسوری یک جبر شرکت‌پذیر جابه‌جایی و یک سوچرجبر لی Universal Central Extension of the Tensor Algebra of a Lie Superalgebra and a Commutative Associative Algebra جبر alg علمی پژوهشی بنیادی S   بررسی نمایش‌های توسیع‌های مرکزی سوپرجبرهای لی به‌علت کاربردشان در بررسی رفتار سیستم‌های فیزیکی همواره مورد علاقۀ ریاضی‌دانان و هم‌چنین فیزیک‌دانان بوده است. در این راستا دست‌یابی به توسیع‌های مرکزی سوپرجبرهای لی بسیار مهم است و اولین سوال در این زمینه، یافتن توسیع‌های مرکزی جامع برای سوپرجبرهای لی است. بررسی توسیع مرکزی جامع جبرهای <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > ، به‌ازای یک جبر شرکت‌پذیر جابه‌جایی یک‌دار <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.png" >  و یک جبر لی با بعد متناهی ساده <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image003.png" > ، در سال 1984 انجام گرفت. پس از آن در سال2011، توسیع مرکزی جامع برای <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.png" > ، برای حالتی که <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image003.png" >  یک سوپرجبر لی بعد متناهی کلاسیک پایه‌ای است، بررسی شد. در این مقاله توسیع مرکزی جامع را برای کلاسی از سوپرجبرهای به فرم <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image005.png" >  بررسی می‌کنیم؛ این کلاس، (سوپر)جبرهایی که در بالا به آن اشاره شد را در بر دارد. روش به‌کار گرفته شده در این مقاله، کاملاً متفاوت از روش‌های قبلی است و به‌علاوه نتایج آن‌ها را پوشش می‌دهد. Introduction Representation as well as central extension are two of the most important concepts in the theory of Lie (super)algebras. Apart from the interest of mathematicians, the attention of physicist are also drawn to these two subjects because of the significant amount of their applications in Physics. In fact, for physicists, the study of projective representations of Lie (super)algebras  are very important. Projective representations of a Lie superalgebra <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="9" > are representations of the central extensions of<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="15" >. So the study of projective representations has two steps; at first, one needs to know the central extensions and then to study their representations.  The first question in the study of central extensions is finding the universal one (if it exists). In 1984, universal central extensions of the algebras of the form <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="56" > for a unital commutative associative algebra <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="9" > and a simple finite dimensional Lie algebra <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" >, were identified. Then in 2011, the case when <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > is a basic classical simple Lie superalgebra was studied by K. Iohara and Y. Koga.  They first study the case for Lie superalgebras <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > of rank 1; then they study <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="9" >-forms of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > and prove the existence of a Chevalley base type for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="8" > using its structure as a basic classical simple Lie superalgebra. This in particular helps them to define an even nondegenerate symmetric invariant bilinear form on  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="8" > Material and methods In this work, we study universal central extensions of Lie superalgebras of the form <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="27" >,  where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > is a finite dimensional  perfect  Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form which is invariant under all  derivations and  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="9" > is a unital commutative associative algebra. Our techniques are totally different from the ones done before; in fact to get our results we use the materials of the previous work of the author (joint with Karl-Hermann Neeb) regarding central extensions of  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="30" > Results and discussion We find the universal central extensions of Lie superalgebras of the form <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="27" >,  where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > is a finite dimensional  perfect  Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form which is invariant under all derivations and  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="9" > is a unital commutative associative algebra.  Conclusion Universal central extensions of Lie superalgebras of the form A &otimes; <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > as above are identified. Our main result covers the results of the previous works in this regard and moreover, since odd nondegenerate invariant bilinear forms on  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > are allowed, we get something more, e.g., the uinversal central extension of A &otimes; <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > for the queer Lie superalgebra  <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="5" > = <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="7" >(n) is also covered by our main theorem.   <a href="./files/site1/files/71/16.pdf">./files/site1/files/71/16.pdf</a> سوپرجبرهای جریانی, توسیع مرکزی, دوهم‌دور, توسیع مرکزی جامع Current superalgebra, 2-cocycle, Central extension, Universal central extension 165 176 http://mmr.khu.ac.ir/browse.php?a_code=A-10-487-1&slc_lang=fa&sid=1 Malihe Yousofzadeh ملیحه یوسف‌زاده ma.yousofzadeh@ipm.ir 10031947532846004458 10031947532846004458 Yes Isfahan University دانشگاه اصفهان، گروه ریاضی