On Jordan doubles of slow growth of Lie superalgebras
On Jordan doubles of slow growth of Lie superalgebras
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DOI：
10.1007/s40863-019-00122-x
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发表时间：
2018-06
期刊：
影响因子：
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通讯作者：
V. Petrogradsky;I. Shestakov
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文献类型：
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作者：
V. Petrogradsky;I. Shestakov
对于任意的李超代数L，我们把它的Jordan double联系起来，它是一个Jordan超代数。这一概念以前由第二作者介绍过（Shestakov in Sib Adv Math 9（2）：83 - 99，1999）。现在我们来研究这个结构的进一步应用。首先，我们证明了Jordan超代数的Gelfand-Kirillov维数可以是任意的。因此，与结合代数和Jordan代数不同（Krause和Lenagan in Growth of algebras and Gelfand-Kirillov dimension，AMS，普罗维登斯，2000; Martinez和Zelmanov in J Algebra 180（1）：211 - 238，1996），Jordan超代数的Gelfand-Kirillov dimension没有Bergman间隙（1，2）的类似物。其次，使用de Morais Costa和Petrogradsky的李超代数（J Algebra 504：291 - 335，2018），我们构造了一个零细阶的Jordan超代数（而且，分量至多是一维的），场的特征不为2。这个例子与特征为零的李代数（Martinez和Zelmanov in Adv Math 147（2）：328 - 344，1999）和特征为非2的Jordan代数（Zelmanov，E.，私人通信）。也就是说，它是无限的，但不是遗传的，只是无限的。在Petrogradsky和Shestakov（Fractal nil graded Lie，associative，poisson，and Jordan superalgebras.）之前，也构造了一个类似的多项式增长缓慢的Jordan超代数。 1804.08441 ，2018）。本例的优点是它是有限宽度4的线性增长，也就是说，它在生成元中的等级具有维度分量，并且这些维度的序列是非周期性的。第三，我们回顾了Petrogradsky和Shestakov（2018）的Poisson和Jordan超代数的构造，从Petrogradsky（J Algebra 466：229 - 283，2016）中引入的另一个李超代数的例子开始。我们讨论了李超代数、结合超代数、泊松超代数和乔丹超代数的自相似性概念。我们还建议的情况下，约旦超代数的圈积的概念。
To an arbitrary Lie superalgebraLwe associate its Jordan double, which is a Jordan superalgebra. This notion was introduced by the second author before (Shestakov in Sib Adv Math 9(2):83–99, 1999). Now we study further applications of this construction. First, we show that the Gelfand–Kirillov dimension of a Jordan superalgebra can be an arbitrary number. Thus, unlike the associative and Jordan algebras (Krause and Lenagan in Growth of algebras and Gelfand–Kirillov dimension, AMS, Providence, 2000; Martinez and Zelmanov in J Algebra 180(1):211–238, 1996), one hasn’t an analogue of Bergman’s gap (1, 2) for the Gelfand–Kirillov dimension of Jordan superalgebras. Second, using the Lie superalgebraof de Morais Costa and Petrogradsky (J Algebra 504:291–335, 2018), we construct a Jordan superalgebrathat is nil finely-graded (moreover, the components are at most one-dimensional), the field being of characteristic not 2. This example is in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk and Gupta–Sidki groups) of Lie algebras in characteristic zero (Martinez and Zelmanov in Adv Math 147(2):328–344, 1999) and Jordan algebras in characteristic not 2 (Zelmanov, E., A private communication). Also,is just infinite but not hereditary just infinite. A similar Jordan superalgebra of slow polynomial growth was constructed before Petrogradsky and Shestakov (Fractal nil graded Lie, associative, poisson, and Jordan superalgebras. arXiv:1804.08441 , 2018). The virtue of the present example is that it is of linear growth, of finite width 4, namely, its-gradation by degree in the generators has components of dimensions, and the sequence of these dimensions is non-periodic. Third, we review constructions of Poisson and Jordan superalgebras of Petrogradsky and Shestakov (2018) starting with another example of a Lie superalgebra introduced in Petrogradsky (J Algebra 466:229–283, 2016). We discuss the notion of self-similarity for Lie, associative, Poisson, and Jordan superalgebras. We also suggest the notion of a wreath product in case of Jordan superalgebras.