Mathematisches Institut SS 2002 der Universität München Set 11 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(41) Let (M^{∗}, ev) be a left dual for M. Show that there is a unique morphism db : I ⟶ M ⊗ M^{∗} satisfying the conditions

(M ⟶^{db ⊗1} M ⊗ M^{∗} ⊗ M ⟶^{1⊗ev} M) = 1_{M}

(M^{∗} ⟶^{1⊗db} M^{∗} ⊗ M ⊗ M^{∗} ⟶^{ev ⊗1} M^{∗}) = 1_{M^{∗}}.

(Uniqueness of the dual basis.)

(42) Let B be the bialgebra 𝕂⟨x, y⟩/I, where I is generated by x^{2}, xy+yx, with the

diagonal Δ(y) = y⊗y, Δ(x) = x⊗1+y⊗x, and the counit ε(y) = 1, ε(x) = 0. A chain complex has the form

M = (… ⟶^{∂_{3}} M_{2} ⟶^{∂_{2}} M_{1} ⟶^{∂_{1}} M_{0})

with ∂_{n−1}∂_{n} = 0.

Show that the category 𝕂-Comp of chain complexes is equivalent to the category B-Comod of B-comodules.

Use the following construction. If M is a chain complex then define a Bcomodule on M = ⊕_{i∈ℕ}M_{i} with the structure map δ : M ⟶ B ⊗ M, δ(m) := y^{i} ⊗ m + xy^{i−1} ⊗ ∂_{i}(m) for all m ∈ M_{i} and for all i ∈ ℕ resp. δ(m) := 1 ⊗ m for m ∈ M_{0}. Conversely if M, δ : M ⟶ B ⊗ M is a B-comodule then we define 𝕂-modules M_{i} := {m ∈ M|∃m′ ∈ M[δ(m) = y^{i} ⊗ m + xy^{i−1} ⊗ m′]} and 𝕂-linear maps ∂_{i} : M_{i} ⟶ M_{i−1} by ∂_{i}(m) := m′ for δ(m) = y^{i} ⊗m+xy^{i−1} ⊗m′. Check that this defines an equivalence of categories.

(Hint: Let m ∈ M ∈ B-Comod . Since y^{i}, xy^{i} form a basis of B we have δ(m) = ∑_{i} y^{i} ⊗ m_{i} + ∑_{i} xy^{i} ⊗ m′_{i}. We apply to this the equation (1 ⊗ δ)δ = (Δ ⊗ 1)δ and compare coefficients to get

δ(m_{i}) = y^{i} ⊗ m_{i} + xy^{i−1} ⊗ m′_{i−1}, δ(m′_{i}) = y^{i} ⊗ m′_{i}

for all i ∈ ℕ_{0} (with m′_{−1} = 0). Consequently for each m_{i} ∈ M_{i} there is exactly one ∂(m_{i}) = m′_{i−1} ∈ M such that

δ(m_{i}) = y^{i} ⊗ m_{i} + xy^{i−1} ⊗ ∂(m_{i}).

Since δ(m′_{i−1}) = y^{i−1} ⊗ m′_{i−1} for all i ∈ ℕ we see that ∂(m_{i}) ∈ M_{i−1}. So we have defined ∂ : M_{i} ⟶ M_{i−1}. Furthermore we see from this equation that ∂^{2}(m_{i}) = 0 for all i ∈ ℕ. Hence we have obtained a chain complex from (M, δ).

If we apply (ϵ ⊗ 1)δ(m) = m then we get m = ∑ m_{i} with m_{i} ∈ M_{i} hence M = ⨁_{i∈ℕ} M_{i}. This together with the inverse construction leads to the required equivalence.)

(43) Find an example of an object M in a monoidal category C that has a left dual but no right dual.

(44) (a) In the category of ℕ-graded vector spaces determine all objects M that

have a left dual.

(b) In the category of chain complexes 𝕂-Comp determine all objects M that have a left dual.

(c) In the category of cochain complexes 𝕂-Cocomp determine all objects M that have a left dual.

Due date: Tuesday, 09.07.2002, 16:15 in Lecture Hall E41