Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

中文翻译：

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弱多项式恒等式的余维增长和 PI 指数的非积分性

让ķ是特征为零的场。在本文中，我们研究了李代数表示的多项式恒等式，也称为弱恒等式或对的恒等式。这些身份由成对的形式决定（一种,大号） 在哪里一种是李代数的结合包络代数大号. 然后是 (一种,大号) （或代表的身份大号关联到一种) 是一个关联多项式，当对大号⊆一种. Kemer 提出的理论是 PI 代数领域最有影响力的成果之一。关联代数的格拉斯曼包络的构造以及代数的恒等式与其格拉斯曼包络的密切关系在其中发挥了关键作用。在这里，我们考虑各种对。我们证明，在某些限制条件下，人们可以发展出一种类似于 Kemer 的研究李代数表示恒等式的理论。因此，我们确定在这种情况下ķ是代数封闭的，如果各种对不包含对应于表示的对sl2(ķ)，并且如果多样性是由关联代数为 PI 的一对生成的，那么它是可溶的。作为用于获得上述结果的方法的另一个结果，并应用 Giambruno 和 Zaicev 论文中的想法，我们能够构建一对 (一种,大号) 使其 PI 指数（如果存在）不能是整数。我们记得 PI 指数存在并且是一个整数一种是结合式（Giambruno 和 Zaicev 的定理），或有限维李代数 (Zaicev)。Gordienko 还证明了 PI 指数存在并且是李代数有限维表示的整数。