Read e-book online An introduction to Hall algebras PDF

By Sjoerd Beentjes

Show description

Read or Download An introduction to Hall algebras PDF

Best geometry books

Geometric Transformations I - download pdf or read online

Virtually everyoneis accustomed to airplane Euclidean geometry because it is mostly taught in highschool. This ebook introduces the reader to a very assorted approach of frequent geometrical evidence. it truly is excited by ameliorations of the airplane that don't modify the sizes and styles of geometric figures.

Additional resources for An introduction to Hall algebras

Example text

18) between Hall algebras. Note that addition is formal, not induced by direct summation in A. In order to have G∗ be a morphism of algebras or coalgebras, something more is required. Let [M ], [N ] ∈ Iso(A), then we see that G∗ ([M ] · [N ]) = M, N A m R FM,N [G (R)], [R]∈Iso(A) whilst G∗ ([M ]) · G∗ ([N ]) = G (M ), G (N ) B m FGS(M ),G (N ) [S]. [S]∈Iso(B) Therefore these two expressions are equal if G preserves the Euler-form and if it preserves the R sets FM,N for all objects M, N, R of A.

Fα1 πt(α1 ) (m) and by extending k-linearly. So the action of a path projects down the element m to the vector space at the starting vertex of the path, chases the element through all consecutive arrows of the quiver, then includes the result back in F M . Given a morphism θ : M → N of representations of Q, we define F θ = i∈Q0 θi . This is a k Q-module morphism since for α ∈ Q1 , m ∈ F M we find M F θ(α · m) = ιN i θi fα πi (m) M = ιN i gα θi πi (m) i∈Q0 = α · F θ(m). i∈Q0 One verifies that the association M → F (M ) defines a covariant functor Rep k (Q) → Mod k Q.

DimC (h) = 2s − rk(A); 3. αi (hj ) = aji for all i, j = 1, 2, . . , s. ∨ Remark. Two such realisations (h1 , Π1 , Π∨ 1 ), (h2 , Π2 , Π2 ) are called isomorphic if there exists ∨ ∗ a complex vector space isomorphism φ : h1 → h2 such that φ(Π∨ 1 ) = Π2 and φ (Π2 ) = Π1 . By [33, Prop. 1], any two realisations of A are isomorphic. 3. Example Take as quiver Q = • → • = A2 , of which the undirected graph is the Dynkin diagram of the simple Lie algebra sl3 (C). Its associated Cartan matrix is 2 −1 .

Download PDF sample

An introduction to Hall algebras by Sjoerd Beentjes


by Charles
4.4

Rated 4.81 of 5 – based on 43 votes