Li J., Su Y., Zhu L.'s 2-Cocycles of original deformative Schrodinger-Virasoro PDF

By Li J., Su Y., Zhu L.

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8. 5. 1. Let T be a triangulated category. 5. TRIANGULATED SUBCATEGORIES 61 triangle X −−−−→ Y −−−−→ Z −−−−→ ΣX such that the objects X and Y are in S, the object Z is also in S. 2. From [TR2] we easily deduce that if S is a triangulated subcategory of T and X −−−−→ Y −−−−→ Z −−−−→ ΣX is a triangle in T, then if any two of the objects X, Y or Z are in S, so is the third. 3. Let T be a triangulated category, S a triangulated subcategory. We define a collection of morphisms M orS ⊂ T by the following rule.

It is the diagram 0 B @ −v g Y ⊕X 0 u 1 0 C A B @ ✲ −w h 0 v 1 0 C A B @ ✲ Z ⊕Y −Σu Σf ΣX ⊕ Z 0 w ✲ 1 C A ΣY ⊕ ΣX . This new candidate triangle is called the mapping cone on a map of candidate triangles. 2. Two maps of candidate triangles u v w X −−−−→ Y −−−−→ Z −−−−→ ΣX         g Σf f h u X −−−−→ Y v w v w −−−−→ Z −−−−→ ΣX and u f X −−−−→ Y −−−−→ Z −−−−→ ΣX         g Σf h u X −−−−→ Y v w −−−−→ Z −−−−→ ΣX are called homotopic if they differ by a homotopy; that is, if there exist Θ, Φ and Ψ below X X u ✲ Θ ✠ u ✲ v Y Y ✲ w Z Φ ✠ v ✲ Z ✲ ΣX Ψ ✠ w ✲ ΣX , with f − f = Θu + Σ−1 {w Ψ} g − g = Φv + u Θ h − h = Ψw + v Φ.

By [TR0], so is 1 X −−−−→ X −−−−→ 0 −−−−→ ΣX. 1, we learn that so is the direct sum 0 X −−−−→ X ⊕ Z −−−−→ Z −−−−→ ΣX. 20. ✷ It is often necessary to know not only that Y is isomorphic to X ⊕ Z, but also to give an explicit isomorphism. 8. Let us be given a triangle u v w X −−−−→ Y −−−−→ Z −−−−→ ΣX. 2. 20 composes to the identity on Z, then the map of triangles X −−−−→ X ⊕ Z −−−−→ Z −−−−→ ΣX         1 1 1 u v u X −−−−→ Y v w −−−−→ Z −−−−→ ΣX is an isomorphism. Proof: First let us establish that there is a map of triangles 0 X −−−−→ X ⊕ Z −−−−→ Z −−−−→ ΣX         1 1 1 u v u X −−−−→ Y v w −−−−→ Z −−−−→ ΣX.

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2-Cocycles of original deformative Schrodinger-Virasoro algebras by Li J., Su Y., Zhu L.


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