Quantum Multilinear Algebra
Quantum Multilinear Algebra
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DOI：
10.2748/tmj/1178227246
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发表时间：
1992-12
影响因子：
0.5
通讯作者：
M. Hashimoto;T. Hayashi
中科院分区：
文献类型：
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作者：
M. Hashimoto;T. Hayashi
基于Jimbo对Yang-Baxter方程Λ# 1的解，我们构造了多元线性代数理论的量子化版本。利用这一点，我们讨论了量子一般线性群的多项式表示。1.内容简介Yang-Baxter运营商473。与Yang-Baxter对相关的代数[j]。量子矩阵477对称代数、外代数和融合过程480除幂代数486 6。Schur模块和Weyl模块Weyl模块的通用性域上GLq的多项式表示。量子行列式和矫直公式515参考文献519引言。量子群是研究量子逆散射方法，特别是Yang-Baxter方程而产生的数学对象。它们是非常显著的Hopf代数，可以看作是Kac-Moody包络代数或李群坐标环的{/-类似物。它们不仅为表示理论增加了新的方面，而且给非交换几何带来了显著的进步，即发现了许多新的例子，如量子线性代数群、量子球等。在本文中，我们研究了一些线性代数对象的量子类似物，如矩阵、对称张量和交变张量以及行列式。我们从Atfl型Yang-Baxter （YB）方程的Jimbo解出发构造了这些方程。并通过我们称之为杨-巴克斯特底线代数的概念来研究它们的结构。作为应用，我们给出了量子一般线性群GLq(N)的Weyl模KλV及其对偶模（Schur模）的实现和自由基，并给出了KλV不可约的判据。我们也给出了量子矩阵双代数的矫直公式的一个类比。我们要强调的是，这些对象定义在任何交换环R和任何单位元素qer上，并且与扩展1991数学主题分类兼容。主要16 w30。472 M. HASHIMOTO和T. HAYASHI的基环r。由此，我们得到了定义在Z[Q， β"]'上的量子一般群的表示理论，其中Q表示不定式。在第1节中，我们介绍了对YB算子的操作（即YB方程的解），称为乘积x，对偶和融合过程。在第2节中，我们将YB算子关联为两个代数，我们称之为对称代数和外部代数。在第3节中，我们将这些工具应用于双代数SE的构造，这些双代数被称为量子矩阵双代数（参见[12]）。在第4节中，我们在Jimbo的aft λ型YB算子的对称代数和外代数上构造了两个YB算子φ和ψ。J使用融合程序。利用YB算子φ，在这些代数的张量积中引入了异常代数结构，并证明了这些代数具有一些我们称之为YB双代数的结构。虽然yb双代数具有代数和协代数的结构，但它不一定是通常意义上的双代数。这些yb双代数的交换性和协交换性是用yb算子ψ来描述的。在第5节中，我们讨论了分级多线性双代数理论的一个类比。回想一下，分幂代数在无特征表示理论的研究中是很重要的（参见[2]，[3],[4]）。这里我们引入第4节中处理过的YB对的分幂代数。它们被定义为第4节中定义的对称代数的分级对偶。这个概念使我们能够以一种自然的方式定义Weyl模块和Schur代数，而无需假设q不是单位的根。从第6节到第9节，我们的兴趣集中在一般线性群的量子变形的表示理论的研究上。在第6节中，我们使用第4节中定义的对称代数和外部代数的yb双代数结构，定义了与一个划分λ相关的（变形的）Weyl模KλV和Schur模LλV。我们证明了LλV和KλV是有限自由/^-模，并且它们是在Z[Q， β ']上定义的，即它们与基扩展兼容。这种性质是所谓普遍自由的类似物。对于这个结果的原始版本，我们建议读者参阅[4]。虽然我们对Lλ v的定义不同于Taft-Towber[37]中的Lλ q(B)，但它们将被证明是等价的（参见命题9.7）。虽然我们在本节中的构造和论证只不过是[4，第二章]中那些构造和论证的变形版本，但我们包括了一些细节，因为它们似乎不是那么明显。在第7节中，我们用我们的语言介绍了舒尔代数、权重和逆变对偶函子的变形版本。我们将证明（变形的）Weyl模是定理7.12意义上的泛权最高模。在第8节中，我们研究了一个基本域K，并讨论了SΈ-comodules的不可约性和完全可约性。本节将介绍正式字符的概念。定理8.9是Weyl模块不可约的钩子长度判据的^-类比。在证明中，杨对称子的Gyoja ^-类似物[14]发挥了重要作用。在第9节中，我们用量子脱量子多元线性代数473终止子证明了拉直公式的^-类似式。即，我们证明了SE v的度k分量SkE v允许SE v -子双模的过滤，其相关的分级对象为©\λ\=k(k λ v)*LχV，其中λ为λ的转置。这个公式是在q= 1的情况下由Doubilet-Rota-Stein[11]提出的。我们的方法与b[4]中的处理方法类似。最后，我们评论了我们的结构与量子包络代数UqQl(N)之间的关系，或者更确切地说，是“量子超代数”，如Lusztig[23]中定义的。由于这些Hopf代数满足定理3.3中的条件，因此在Jimbo的Atfl型YB对上的量子矩阵双代数之间存在双代数对。X和量子超代数。因此，由于双代数的一般理论，量子超代数作用于我们的Schur和Weyl模。此外，当它们被视为量子超代数上的表示时，不可约准则（定理8.9）也是有效的。本论文第一版投稿后，作者收到了DipperJames[10]、Noumi-Yamada-Mimachi[29]、parshll - wang[30]的预印本，这些预印本与我们的论文有部分重合。我们特别感谢星野光夫教授、菅井幸弘教授、松村秀之教授和土屋昭弘教授提供的宝贵意见。1. Yang-Baxter运营商。设V是自由的/^-模。V上的一个Yang-Baxter（或YB）算子是一个自同构βveEndR(V®V}，使得（1.1）（βV）l°（βV）2°(βvϊl=(βV)2°（βV） l°（βV）2，一个Yang-Baxter对V=(V9 βV)是一个自由/^-模V，在V上配置了一个Yang-Baxter算子βV。例(1)（琐碎的扭曲）。设K为自由模。则由τv(u®u') = u l®u定义的映射τv: = τv Y是v上的YB算子。我们称τv为v上的平凡扭转算子。设K是一个自由的/^-模，其基为{ui9 u2，…， n}，设q是r的可逆元素，则Jimbo的A^_ l型YB算子是由以下公式定义的映射：我们称{wj为（F, βv）的标准基。这个算子也满足Iwahori的二次方程
We construct a quantized version of the theory of multilinear algebra, based on Jimbo's solution of Yang-Baxter equation of type Λ#l 1. Using this, we discuss the polynomial representations of quantum general linear groups. CONTENTS Introduction 471 1. Yang-Baxter operators 473 2. Algebras associated with Yang-Baxter pairs 476 3. Quantum matrices 477 4. Symmetric algebras, exterior algebras and the fusion procedure 480 5. Divided power algebra 486 6. Schur modules and Weyl modules 493 7. Universality of Weyl modules 505 8. Polynomial representation of GLq over a field 510 9. Quantum determinants and straightening formulas 515 References 519 Introduction. Quantum groups are mathematical objects which arose from the study of the quantum inverse scattering method, especially the Yang-Baxter equation. They are very remarkable Hopf algebras and can be considered as {/-analogues of Kac-Moody enveloping algebras or of coordinate rings of Lie groups. Not only have they added new aspects to representation theory, but also they have brought to non-commutative geometry a remarkable progress, i.e. the discovery of many new examples such as quantum linear algebraic groups, quantum spheres and so on. In this article, we study quantum analogues of some linear-algebraic objects such as matrices, symmetric and alternating tensors, and determinants. We construct these from Jimbo's solution of Yang-Baxter (YB) equation of type Atfl. x and investigate their structure via the notion which we call Yang-Baxter bίalgebras. As applications, we give realizations and free bases of Weyl modules KλVand their dual modules (Schur modules) of quantum general linear groups GLq(N), and give a criterion for the irreducibility of KλV. We also give an analogue of the straightening formula for quantum matric bialgebras. We would like to emphasize that these objects are defined over any commutative ring R and any unit element q e R and are compatible with extensions 1991 Mathematical Subject Classification. Primary 16W30. 472 M. HASHIMOTO AND T. HAYASHI of the base ring R. Hence, we can get the representation theory of quantum general group 'defined over Z[Q, β"]', where Q denotes an indeterminate. In Section 1, we introduce operations on YB operators (i.e. solutions of the YB equation) called the product x , dual , and fusion procedure. In Section 2, we associate with a YB operator two algebras which we call the symmetric and the exterior algebras. In Section 3, we apply these tools to the construction of bialgebras SE which are called quantum matric bialgebras (cf. [12]). In Section 4, we construct two YB operators φ and ψ on the symmetric and exterior algebras of Jimbo's YB operators of type Aftϊ. j using the fusion procedure. With the YB operator φ, we introduce unusual algebra structures into tensor products of these algebras, and prove that these algebras have some structures which we call YB-bialgebras. Though a YB-bialgebra has structures of an algebra and a coalgebra, it is not necessarily a bialgebra in the usual sense. The 'commutativity' and the 'cocommutativity' of these YB-bialgebras are described in terms of the YB-operator ψ. In Section 5, we discuss an analogue of the theory of graded multilinear bialgebras. Recall that divided power algebras have been important in the study of characteristicfree representation theory (cf. [2], [3], [4]). Here we introduce divided power algebras of the YB pairs treated in Section 4. They are defined to be graded duals of symmetric algebras defined in Section 4. This concept enables us to define Weyl modules and Schur algebras in a natural way without assuming that q is not a root of unity. From Sections 6 to 9, our interest is concentrated on the study of the representation theory of quantum deformations of general linear groups. In Section 6, we define (deformed) Weyl modules KλV and Schur modules LλV associated to a partition λ, using the YB-bialgebra structure of the symmetric and exterior algebras defined in Section 4. We prove that LλV and KλV are finite free /^-modules and that they are 'defined over Z[Q, β"]' in the sense that they are compatible with base extensions. This property is an analogue of the so-called universal freeness. For the original version of this result, we refer the reader to [4]. Though our definition of LλV is different from L λ q(B) in Taft-Towber [37], they will turn out to be equivalent (cf. Proposition 9.7). Though our construction and argument in this section are nothing but the deformed versions of those in [4, Chapter II], we include some details, since they do not seem to be so obvious. In Section 7, we introduce the deformed versions of the Schur algebra, weights, and the contravariant dual functor in our language. We will show that (deformed) Weyl modules are universal highest weight modules in the sense of Theorem 7.12. In Section 8, we work over a base field K, and discuss the irreducibility and complete reducibility of SΈ-comodules. The notion of formal characters is introduced in this section. Theorem 8.9 is a ^-analogue of the hook length criterion for the irreducibility of Weyl modules. In the proof, Gyoja's ^-analogues of Young symmetrizers [14] play important roles. In Section 9, we prove a ^-analogue of the straightening formula using quantum deQUANTUM MULTILINEAR ALGEBRA 473 terminants. Namely, we prove that the degree k component SkE v of SE v admits a filtration of SE v -subbicomodules whose associated graded object is ©\λ\=k(KλV)*LχV, where λ is the transpose of λ. This formula was originated by Doubilet-Rota-Stein [11] in the case q=l. Our approach is a ^-analogue of the treatment in [4]. Lastly, we remark on the relation between our construction and the quantum enveloping algebra UqQl(N), or rather the "quantum hyperalgebra" such as that defined in Lusztig [23]. Since these Hopf algebras satisfy the conditions in Theorem 3.3, there are pairings of bialgebras between the quantum matrix bialgebra over Jimbo's YB pair of type Atfl. x and the quantum hyperalgebra. Hence, thanks to the general theory of bialgebras, the quantum hyperalgebra acts on our Schur and Weyl modules. Moreover, the criterion for the irreducibility (Theorem 8.9) is also valid when they are viewed as representations over the quantum hyperalgebra. After the submission of the first version of this work, the authors received preprints by DipperJames [10], Noumi-Yamada-Mimachi [29], Parshall-Wang [30], which have some overlap with our paper. Our special thanks are due to Professors Mitsuo Hoshino, Yukihiro Kanie, Hideyuki Matsumura and Akihiro Tsuchiya for valuable advice. 1. Yang-Baxter operators. Let V be a free /^-module. A Yang-Baxter (or YB) operator on V is an automorphism βveEndR(V® V} such that (1.1) (βV)l°(βv)2°(βvϊl=(βv)2°(βv)l°(βV)2, A Yang-Baxter pair V=(V9 βv) is a free /^-module V equipped with a Yang-Baxter operator βv on V. Here we give some examples of Yang-Baxter pairs. EXAMPLE (1) (trivial twisting). Let K be a free ^-module. Then the map τv : = τv Y defined by τv(u®u') = u l ®u is a YB operator on V. We call τv the trivial twisting on V. EXAMPLE (2) (Jimbo's operators of type A^l.^. Let K be a free /^-module with a basis {ui9 u2, . . . , UN} and let q be an invertible element of R. Then Jimbo's YB operators of type A^_ l is the map defined by the following formula: (1.2) (ί >j} . We call {wj the standard basis of (F, βv). This operator also satisfies Iwahori's quadratic equation