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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
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> 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 </span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image001.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">، بهازای یک جبر شرکتپذیر جابهجایی یکدار </span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image002.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> و یک جبر لی با بعد متناهی ساده </span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image003.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">، در سال 1984 انجام گرفت. پس از آن در سال2011، توسیع مرکزی جامع برای </span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image004.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">، برای حالتی که </span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image003.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> یک سوپرجبر لی بعد متناهی کلاسیک پایهای است، بررسی شد. در این مقاله توسیع مرکزی جامع را برای کلاسی از سوپرجبرهای به فرم </span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image005.png" > </span></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> 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.</span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;"></span></span></span><br>
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"></span></span>
<strong>Introduction</strong><br>
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.<br>
Projective representations of a Lie superalgebra <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image002.gif" width="9" > are representations of the central extensions of<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_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. <br>
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:Users1AppDataLocalTempmsohtmlclip1 1clip_image006.gif" width="56" > for a unital commutative associative algebra <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image008.gif" width="9" > and a simple finite dimensional Lie algebra <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image010.gif" width="5" >, were identified. Then in 2011, the case when <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_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:Users1AppDataLocalTempmsohtmlclip1 1clip_image010.gif" width="5" > of rank 1; then they study <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image012.gif" width="9" >-forms of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image010.gif" width="5" > and prove the existence of a Chevalley base type for <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_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:Users1AppDataLocalTempmsohtmlclip1 1clip_image016.gif" width="8" ><br>
<strong>Material and methods</strong><br>
In this work, we study universal central extensions of Lie superalgebras of the form <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image018.gif" width="27" >, where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_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:Users1AppDataLocalTempmsohtmlclip1 1clip_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:Users1AppDataLocalTempmsohtmlclip1 1clip_image020.gif" width="30" ><br>
<strong>Results and discussion</strong><br>
We find the universal central extensions of Lie superalgebras of the form <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image018.gif" width="27" >, where <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_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:Users1AppDataLocalTempmsohtmlclip1 1clip_image008.gif" width="9" > is a unital commutative associative algebra. <br>
<strong>Conclusion</strong><br>
Universal central extensions of Lie superalgebras of the form A ⊗ <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_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:Users1AppDataLocalTempmsohtmlclip1 1clip_image010.gif" width="5" > are allowed, we get something more, e.g., the uinversal central extension of A ⊗ <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image010.gif" width="5" > for the queer Lie superalgebra <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image010.gif" width="5" > = <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip1 1clip_image022.gif" width="7" >(n) is also covered by our main theorem. <br>
<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