论文标题
倾斜理论I：基础的粉丝和多面有
Fans and polytopes in tilting theory I: Foundations
论文作者
论文摘要
对于一个有限维数代数$ a $在字段$ k $上，$ a $的2项淤积络合物可提供一个简单的复合物$δ（a）$，称为$ g $ $ $ simplicial complect。我们给出$ H $ - 向量和Dehn-Sommerville方程$δ（a）$的倾斜理论解释。使用$ g $ - 向量的2-级淤积络合物，$δ（a）$在真正的grothendieck组中提供了非发明的粉丝$σ（a）$ $σ（a）$ $ k_0（\ mathsf {proj} a）_ {\ mathbb {rsbb {r}} $称为$ g $ fan。我们提供了$σ（a）$的几个基本属性，包括标志固定，符号分解，diDempotent降低，Jasso降低，成对阳性以及与$ A $ A $ A-Modules的Newton Polytopes的连接。此外，$σ（a）$给出了$ k_0（\ mathsf {proj} a）_ {\ mathbb {r}} $称为$ g $ -polytope of $ a $ a $ a $ a $ a $ a $ a $。如果$ p（a）$是凸，我们称$ a $ g $ -convex。在这种情况下，我们表明它是一种反身层，并且双重多层由$ a $的两项简单志同道合收藏给出。恰好有7个凸出$ g $ - polyogons to Insomorphism。我们提供了代数的分类，其$ g $ - polytopes是顺畅的。 我们研究了两个重要类别的代数类别的$ g $ - 粉丝和$ g $ - $ - 聚会。我们表明，Coxeter风扇给出了经典或广义前代数的$ G $ fan。它是$ g $ -convex，并且仅当它是类型$ a $ a $ a或$ b $的，在这种情况下，其$ g $ - polytope是短根polytope的双重多层。此外，我们将Brauer图代数分类为$ G $ -CONVEX，并将其$ g $ - polytopes描述为$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ g $ convex。
For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $Δ(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $Δ(A)$. Using $g$-vectors of 2-term silting complexes, $Δ(A)$ gives a nonsingular fan $Σ(A)$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-fan. We give several basic properties of $Σ(A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of $A$-modules. Moreover, $Σ(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. There are precisely 7 convex $g$-polyogons up to isomorphism. We give a classification of algebras whose $g$-polytopes are smooth Fano. We study $g$-fans and $g$-polytopes of two important classes of algebras. We show that the $g$-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or $B$, and in this case, its $g$-polytope is the dual polytope of the short root polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as the root polytopes of type $A$ or $C$.
