On graded Frobenius algebras
On graded Frobenius algebras
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DOI：
10.1016/s0021-8693(03)00262-x
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发表时间：
2003-09
影响因子：
0.9
通讯作者：
Takayoshi Wakamatsu
中科院分区：
文献类型：
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作者：
Takayoshi Wakamatsu
Frobenius代数经常出现在数学中，例如有限群的群代数和限制李代数的限制包络代数，更一般的有限维Hopf代数都是Frobenius代数。最近，关于Artin-Shelter正则代数或量化多项式环，Frobenius代数显得相当自然。详情见Artin和Schelter[1]和Smith[4]。因此，知道如何获得Frobenius代数是很重要的。在前文[5]中，作者证明了在代数闭域上，任何没有半单部分的Frobenius代数都可以由幂零的Frobenius代数构造而成。然而，一般情况下，构造可能的幂零Frobenius代数并非易事。因此，在本文中，我们主要研究分次Frobenius代数，并将其构造为张量代数的因子。设A是代数闭域K上的代数，AXA是双模。对于一个自同构σ∈autK(A)和一个同构γ：AXA∼→σXσ，我们考虑一个线性映射θ：X⊗d→K，其中σXσ表示由σ和⊗均值⊗A构成的AXA的扭曲双模。我们称这个三重Φ=(σ，γ，θ)是分次Frobenius代数的容许系统，如果θ满足θ(x1⊗x2⊗··⊗XD)=θ
Frobenius algebras appear often in mathematics, for example, group algebras of finite groups and restricted enveloping algebras of restricted Lie algebras, more generally, finitedimensional Hopf algebras are all Frobenius. Recently, relating to Artin–Shelter’s regular algebras or quantized polynomial rings, Frobenius algebras appear quite naturally. For details, see Artin and Schelter [1] and Smith [4]. Hence, it is important to know the way of obtaining Frobenius algebras. In the previous paper [5], over an algebraically closed field, the author proved that any Frobenius algebra without semisimple part can be constructed from a nilpotent Frobenius algebra. However, it is not so easy to construct possible nilpotent Frobenius algebras generally. So, in this paper, we concentrate on graded Frobenius algebras and construct them as factors of tensor algebras. Let A be an algebra over an algebraically closed field K and AXA a bimodule. For an automorphism σ∈ AutK (A) and an isomorphism γ: AXA∼→ σ Xσ, we consider a linear map θ: X⊗ d→ K, where σ Xσ stands for the twisted bimodule of AXA by σ and⊗ means⊗ A. We call the triple Φ=(σ, γ, θ) an admissible system for a graded Frobenius algebra if θ satisfies θ (x1⊗ x2⊗···⊗ xd)= θ