Using Lemma 15, we compute. This shows that is a subcoalgebra and induces a coproduct on.
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As a subcoalgebra of a coassociative coalgebra is clearly coassociative itself, becomes a coaugmented coalgebra with the coproduct given by It is immediate that is conilpotent, as it is a subcoalgebra of T V. We claim that is even cocommutative. We then have and therefore. There is a unique homomorphism of coaugmented coalgebras such that. As in the proof of Proposition 17, it suffices to show that there is a unique homomorphism of coalgebras satisfying.
Recall that is the unique coalgebra homomorphism extending f. As the image of is contained in the subspace of of all symmetric elements, which is im N. We obtain an induced homomorphism of coalgebras Similarly, a coalgebra homomorphism gives rise to a coalgebra homomorphism that is uniquely determined by by Proposition As N is injective, this shows uniqueness of.
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The induced homomorphism of coaugmented coalgebras is denoted by and is given by. If M is both a left and a right comodule over C and if the compatibility relation If C and D are coaugmented and if f is a homomorphism of coaugmented coalgebras, the comodule structure is compatible with the counit in the sense that the diagram. Observe that then makes into a comodule over. Given a linear map , one easily checksthat is a coderivation if and only if is. From 13 , 23 and the compatibility with the counit 24 it then follows that.
Hence, d decomposes as. We then call the pair C , d a differential graded coassociative coalgebra DGC for short. A homomorphism of DGCs is then a coalgebra homomorphism that is also a homomorphism of DG vector spaces; homomorphisms of coaugmented DGCs are defined accordingly. From Proposition 20 and Section 2. Let C be a coassociative coalgebra. Both parts of the proposition are straightforward computations which are left to the reader.
Theorem Let D be a cocommutative coaugmented coalgebra and a homomorphism of coaugmented coalgebras. The linear map is then an isomorphism. Together with Proposition 20, this shows. The first part of Theorem 22 actually holds for a broader class of comodules over see for example [ 8 ], Lemma 2.
Remark This yields abijection. The map is a homomorphism of graded algebras. We show by induction over that We compute. Proof of Theorem Let be a coderivation, that is.
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Inductively,we then get. For , let be the linear map defined by. From being a subcoalgebra of T V and 17 , it follows that. We then have. The graded vector space is called the dual space of V. By degree reasons,. Note that and if g is a homogeneous linear map with domain W , we have. We say that V is of finite type if V k is finite-dimensional for all.
Notions as - graded or - graded are defined accordingly. If V is of finite type and if for all the decomposition has only finitely many non-trivial summands, then the canonical inclusion is an isomorphism. It is well-known that for finite-dimensional ungraded vector spaces V 1 , …, V n , the canonical inclusion is an isomorphism.
Corollary If V is of finite type and - graded, the canonical inclusion is an isomorphism. Then If V is of finite type and - graded ,. Under this identification , is the usual multiplication on. While the latter is isomorphic to by Remark 2, the former is isomorphic to via the linear map. This yields an isomorphism. With a similar reasoning as in the proof of Corollary 27, one obtains.
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It is then a straightforward computation to show that is indeed the usual multiplication on. For this, we will mostly follow [ 3 ], although the original references for Section 3.
While the first two equations may be summarized by saying that l 1 is a differential on the non-associative graded algebra L , l 2 , a comparison with 11 shows that the third one describes the defect of the Jacobi identity in L , l 2. These homomorphisms are strict in the sense that they strictly preserve all brackets.
Theorem 32 [ 5 ], Theorem Sketch of proof. Note that for fixed n , we can write 27 as. We have proved the following. Combine the brackets in Proposition 33 to a single element of degree one and compare with The equation 31 can be written more conveniently as See Remark 26 and note that as L is -graded, each of degree one vanishes on by degree reasons.
As commented in [ 8 ], Remark 5. Observe that. Assume that is a homomorphism of coalgebras such that for all n and let. Hence, by Proposition While it is a straightforward computation to show equivalence between these definitions, it is convenient to save this for Section 5. In [ 3 ], representations up to homotopy were described under some finiteness assumptions as differentials on.
A representation up to homotopy of L on V is a linear map of degree one that satisfies We prove the following version of [ 8 ], Theorem 5. One should therefore really think of L being representated on a DG vector space. Let be a DGLA and be the corresponding linear degree one map. This is the case if and only if the linear degree one map defined by satisfies The first part follows immediately from Remark 39 and the explicit construction of see Proposition It is straightforward to check that for homogeneous,.
Proof of Theorem As , a linear degree one map can be decomposed into linear degree one maps and ; the latter being equivalent to the choice of a degree one element. For homogeneous,. As , satisfying 35 is equivalent to 33 holding on. We also have.welduturlitas.tk
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Example 47 The trivial representation on a DG vector space. In particular, there is a trivial representation of L on. The induced representation of L on V [ n ] is given by.