English
If A carries a bialgebra structure over R, then the Laurent polynomial ring A[T; T^{-1}] carries a natural bialgebra structure over R.
Русский
Если A имеет структуру биалгебры над R, то кольцо Лорента A[T; T^{-1}]Naturally inherits a бialgebra structure над R.
LaTeX
$$LaurentPolynomial.instBialgebra : Bialgebra R A[T;T^{-1}]$$
Lean4
instance instBialgebra : Bialgebra R (MonoidAlgebra A M)
where
counit_one := by simp only [one_def, counit_single, Bialgebra.counit_one]
mul_compr₂_counit := by ext; simp
comul_one := by
simp only [one_def, comul_single, Bialgebra.comul_one, Algebra.TensorProduct.one_def, TensorProduct.map_tmul,
lsingle_apply]
mul_compr₂_comul := by
ext a b c d
simp only [Function.comp_apply, LinearMap.coe_comp, LinearMap.compr₂_apply, LinearMap.mul_apply', single_mul_single,
comul_single, Bialgebra.comul_mul, ← (Coalgebra.Repr.arbitrary R b).eq, ← (Coalgebra.Repr.arbitrary R d).eq,
Finset.sum_mul_sum, Algebra.TensorProduct.tmul_mul_tmul, map_sum, TensorProduct.map_tmul, lsingle_apply,
LinearMap.compl₁₂_apply, LinearMap.coeFn_sum, Finset.sum_apply,
Finset.sum_comm (s := (Coalgebra.Repr.arbitrary R b).index)]
-- TODO: Generalise to `MonoidAlgebra A M →ₐc[R] MonoidAlgebra A N` under `Bialgebra R A`
