Mathematical Researches
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کران بالایی برای کلاس پوچتوانی جبرهای لایبنیتز
An upper bound for the nilpotency class of Leibniz algebras
جبر
alg
علمی پژوهشی بنیادی
S
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="font-family:Calibri,sans-serif"><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin"">در این مقاله ،کلاس پوچ­توانی جبر­های لایبنیتز با تولید متناهی را با کلاس پوچ­توانی زیرجبر­های آنها مقایسه می­کنیم و کران بالایی برای آن ارائه می­دهیم. به عنوان نتیجه اصلی نشان می­دهیم که اگر </span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">A</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> یک جبر لایبنیتز پوچ­توان با تولید متناهی و </span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">d > 1</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> تعداد مولد­های کمین آن باشد و </span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">c</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> ماکزیمم کلاس پوچ­توانی زیرجبرهای بیشین </span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">A</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> باشد</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin"">،</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> آن­گاه </span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">A</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> پوچ­توان از کلاس حداکثر </span></span><m:omath><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌊</m:r></span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>cd/(d</m:r></span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>-</m:r></span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1) </m:r></span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌋</m:r></span></span></m:omath><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> است که در آن </span></span><m:omath><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌊</m:r></span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌋</m:r></span></span></m:omath><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> تابع جزء صحیح می­باشد. همچنین با ارائه ساختار یک ­خانواده از جبر­های لایبنیتز نشان می­دهیم که در حالت </span></span><span dir="LTR" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">d = 1</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"Arial",sans-serif">،</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> کلاس پوچ­توانی یک جبر­ لایبنیتز</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"Arial",sans-serif">،</span></span><span lang="FA" style="font-size:10.0pt"><span style="font-family:"B Nazanin""> بیشترین مقدار خود را اختیار می­کند که در واقع بعد آن جبر لایبنیتز است.</span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="direction:rtl"><span style="unicode-bidi:embed"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span lang="FA" style="font-size:10.0pt"><span style="line-height:107%"><span style="font-family:"B Nazanin""></span></span></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Introduction</span></b></span></span></span><br>
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" style="position:absolute; left:0; text-align:left; margin-left:79px; margin-top:113px; width:17.65pt; height:0; z-index:251676672" type="#_x0000_t32"> <v:stroke endarrow="block"> </v:stroke></v:shape><span style="position:absolute; margin-left:78px; margin-top:107px; width:26px"><span style="z-index:251676672"><span style="left:0px"><span style="height:12px"></span></span></span></span><span style="font-size:10.0pt">The notion of Leibniz algebras is introduced by Blokh in 1965 as a noncommutative version of Lie algebras and rediscovered by Loday in 1993. A Leibniz algebra is a vector space <i>A</i> over a field <i>F</i> together with a bilinear map </span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">[ , ] : <i>A</i> × <i>A</i> <i> A</i></span></span><span style="font-size:10.0pt"> usually called the Leibniz bracket of <i>A</i>, satisfying the Leibniz identity:</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="font-family:Calibri,sans-serif"> <span dir="RTL" lang="AR-SA" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> <m:r></m:r></span></span><m:omath><m:r> </m:r></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">[<i>x</i>,[<i>y</i>,<i>z</i>]] = [[<i>x</i>,<i>y</i>],<i>z</i>] – [[<i>x</i>,<i>z</i>],<i>y</i>] <i>, x,y,z </i></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif">∈ </span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">A.</span></span></i> </span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">The classification of nilpotent Leibniz algebras is one of the most important subject in the study of Leibniz algebras. In the present paper, we obtain an upper bound for the nilpotency class of a finitely generated nilpotent Leibniz algebra <i>A</i> in terms of the maximum of the nilpotency classes of maximal subalgebras of <i>A</i> and the minimal number of generators of <i>A</i>. </span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Main results</span></b><span style="font-size:10.0pt"></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">Throughout the paper, any Leibniz algebra <i>A</i> is considered over a fixed field <i>F</i>, <i>c</i> denotes the maximum of the nilpotency classes of maximal subalgebras of <i>A</i>, <i>d</i> is the minimal number of generators of <i>A</i> and </span><m:omath><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌊</m:r></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r> </m:r></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌋</m:r></span></span></m:omath><span style="font-size:10.0pt"> denotes the integral part. Moreover, we inductively define: </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">A</span></span></i><sup><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">1</span></span></sup><span style="font-size:10.0pt">=<i>A</i> and </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">A<sup>n</sup></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> = [<i>A<sup>n</sup></i><sup>-1</sup>,<i>A</i>]</span></span><span style="font-size:10.0pt">, for </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">n</span></span></i> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">≥</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> 2</span></span><span style="font-size:10.0pt">.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Lemma 1. </span></b><span style="font-size:10.0pt">Let <i>A</i> be a nilpotent Leibniz algebra and <i>M</i> be a maximal subalgebra of <i>A</i>. Then <i>M</i> is a two-sided ideal of <i>A</i>.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Lemma 2. </span></b><span style="font-size:10.0pt">Let <i>I </i>be a two-sided ideal of a Leibniz algebra <i>A</i> and </span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">{<i>x</i><sub>1</sub>,<i>x</i><sub>2</sub>, … ,<i>x<sub>k</sub></i>}</span></span> <span style="font-size:10.0pt"><span style="letter-spacing:.5pt">be a subset of <i>A</i> which contains at least <i>n</i> elements of <i>I</i> </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:.5pt">(<i>n</i> ≤ <i>k</i>)</span></span></span><span style="font-size:10.0pt"><span style="letter-spacing:.5pt">. Then </span></span><m:omath><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:.5pt"><m:r> </m:r></span></span></span></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:.5pt">[[[<i>x</i><sub>1</sub>,<i>x</i><sub>2</sub>],<i>x</i><sub>3</sub>], … ,<i>x<sub>k</sub></i>] </span></span></span><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:.5pt">∈</span></span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:.5pt"> <i>I<sup>n</sup></i>.</span></span></span> </span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Theorem 3. </span></b><span style="font-size:10.0pt">Let <i>A</i> be a finitely generated nilpotent Leibniz algebra such that </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">d</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> > 1</span></span><span style="font-size:10.0pt">. Then <i>A</i> is nilpotent of class at most </span><m:omath><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌊</m:r></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>cd</m:r></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>/(</m:r></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>-</m:r></span></span><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1)</m:r></span></span><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌋</m:r></span></span></m:omath><span style="font-size:10.0pt">.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="letter-spacing:.4pt">Corollary 4. </span></span></b><span style="font-size:10.0pt"><span style="letter-spacing:.4pt">Let <i>A</i> be a finitely generated nilpotent Leibniz algebra and </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:.4pt">{<i>x</i><sub>1</sub>,<i>x</i><sub>2</sub>, … ,<i>x<sub>d</sub></i>}</span></span></span><span style="font-size:10.0pt"> be a minimal generating set of <i>A</i> with </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">d</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> > 1</span></span><span style="font-size:10.0pt">. Then</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="font-family:Calibri,sans-serif"><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>cl</m:r></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r><i>A</i></m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></span></span><span dir="RTL" lang="AR-SA" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><m:r>≤</m:r></span></span><m:func><m:funcpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:funcpr><m:fname><m:limlow><m:limlowpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:limlowpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>max</m:r></span></span></i></m:e><m:lim><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>1≤</m:r><m:r>i</m:r><m:r>≤</m:r><m:r>d</m:r></span></span></i></m:lim></m:limlow></m:fname><m:e><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌊</m:r></span></span><m:f><m:fpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:fpr><m:num><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>cl</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:ctrlpr></m:ctrlpr></span></span></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>Mi</m:r></span></span></i></m:e></m:d><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r><m:r><i>d</i></m:r></span></span></m:num><m:den><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>d</m:r></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>-</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></span></span></m:den></m:f><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>⌋</m:r></span></span></m:e></m:func></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">,</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt">where </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Mi</span></span></i><span style="font-size:10.0pt"> is the two-sided ideal generated by the set</span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> { <i>x</i><sub>1</sub> , …, <i>x</i><sub>i-1 </sub>, <i>x</i><sub>i+1 </sub>, … ,<i>x<sub>d</sub></i>}</span></span><span style="font-size:10.0pt"> and <i>cl</i> denotes the nilpotency class.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Corollary 5. </span></b><span style="font-size:10.0pt">Let <i>A</i> be a finitely generated nilpotent Leibniz algebra of class <i>k</i> with </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">d</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> > 1</span></span><span style="font-size:10.0pt">. Then </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">c</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> ≤ <i>k</i> ≤ 2<i>c</i></span></span><span style="font-size:10.0pt">.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Corollary 6. </span></b><span style="font-size:10.0pt">Let <i>A</i> be a finitely generated nilpotent Leibniz algebra such that </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">d</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> > </span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">c</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">+1.</span></span><span style="font-size:10.0pt"> Then <i>A</i> is nilpotent of class <i>c</i>.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Corollary 7. </span></b><span style="font-size:10.0pt">Let <i>A</i> be a finitely generated nilpotent Leibniz algebra of class 2<i>c</i> such that </span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">d</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> > 1</span></span><span style="font-size:10.0pt">. Then <i>d</i>=2.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:23.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt">Proposition 8. </span></b><span style="font-size:10.0pt">There is not any non-Lie Leibniz algebra with at least two generators, whose maximal subalgebras are all abelian.</span> <span style="font-size:10.0pt"></span></span></span></span></span><br>
جبر لایبنیتز , کلاس پوچتوانی , مولد کمین.
Leibniz algebra, nilpotency class, minimal generator.
144
152
http://mmr.khu.ac.ir/browse.php?a_code=A-10-1131-2&slc_lang=fa&sid=1
Hesam
Safa
حسام
صفا
hesam.safa@gmail.com
10031947532846005790
10031947532846005790
Yes
University of Bojnord
دانشگاه بجنورد