By Karakhanyan M.I., Khor'kova T.A.
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The constitution of the legislation in physics is basically according to symmetries. This booklet is on Lie algebras, the math of symmetry. It has grown from lectures for undergraduates in theoretical and mathematical physics and offers an intensive mathematical therapy of finite dimensional Lie algebras and Kac-Moody algebras.
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Additional resources for A Characteristic Property of the Algebra C(Q)B
Compute explicitly: a) the dual coalgebra of K[x] and of Γ(K x), b) the coalgebra structure of K[x], resp Γ(K x), which makes it a bialgebra and which is uniquely determined by ∆(x) = x ⊗ 1 + 1 ⊗ x. c) Compare the results of a) and b). 6. Polynomial algebra continued. e. for the algebra of coinvariants S(V ) = n ((V ⊗n )Sn ) and the algebra of invariants Γ(V ) = n ((V ⊗n )Sn ) over the vector space V . 7. Symmetric algebra as bialgebra. Show that S(V ) is a Hopf algebra for ∆ uniquely determined by ∆(v) = v ⊗ 1 + 1 ⊗ v.
The term ¯ ¯ under d2 ◦d2 comes with the sign s−1 x(1) ⊗s−1 x(2) ⊗s−1 x(3) coming from (∆⊗id) ∆ (−1)p+q (−1)p . Indeed, the first one comes from the application of the first copy of d2 , the second one comes from the application of the second copy of d2 . The term ¯ ∆ ¯ under d2 ◦ d2 comes with the s−1 x(1) ⊗ s−1 x(2) ⊗ s−1 x(3) coming from (id ⊗ ∆) p p−1 q sign (−1) (−1) (−1) . Indeed, the first one comes from the application of the first copy of d2 , the second one comes from the fact that d2 , which is of degree −1, jumps over a variable of degree p − 1, the third one comes from the application of the second copy of d2 .
2 (Twisting morphism fundamental theorem). Let A be a connected wdga algebra and let C be a connected wdga coalgebra. For any twisting morphism α : C → A the following assertions are equivalent: (1) (2) (3) (4) the the the the right twisted tensor product C ⊗α A is acyclic, left twisted tensor product A ⊗α C is acyclic, ∼ dga coalgebra morphism fα : C − → BA is a quasi-isomorphism, ∼ dga algebra morphism gα : ΩC − → A is a quasi-isomorphism. ¯ Proof. Since we require A to be connected, we have A = A⊕K 1, where the elements of the augmentation ideal A¯ have positive degree and positive weight.
A Characteristic Property of the Algebra C(Q)B by Karakhanyan M.I., Khor'kova T.A.