Invariant theory in exterior algebras and Amitsur-Levitzki type theorems

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• 作者：Minoru Itoh ^{itoh@sci.kagoshima-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 15A72 ; 15A75 ; secondary ; 16R ; 15A24 ; 15B33
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：22 January 2016
• 年：2016
• 卷：288
• 期：Complete
• 页码：679-701
• 全文大小：422 K

文摘

This article discusses invariant theories in some exterior algebras, which are closely related to Amitsur–Levitzki type theorems.

First we consider the exterior algebra on the vector space of square matrices of size n, and look at the invariants under conjugations. We see that the algebra of these invariants is isomorphic to the exterior algebra on an n-dimensional vector space. Moreover we give a Cayley–Hamilton type theorem for these invariants (the anticommutative version of the Cayley–Hamilton theorem). This Cayley–Hamilton type theorem can also be regarded as a refinement of the Amitsur–Levitzki theorem.

We discuss two more Amitsur–Levitzki type theorems related to invariant theories in exterior algebras. One is a famous Amitsur–Levitzki type theorem due to Kostant and Rowen, and this is related to O(V)$O(V)$-invariants in Λ(Λ₂(V))$\mathrm{\Lambda }({\mathrm{\Lambda }}_2(V))$. The other is a new Amitsur–Levitzki type theorem, and this is related to GL(V)$\mathit{GL}(V)$-invariants in Λ(Λ₂(V)⊕S₂(V^⁎))$\mathrm{\Lambda }({\mathrm{\Lambda }}_2(V)\oplus S_2(V^{⁎}))$.