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On twisted tensor products of algebras. (English) Zbl 0842.16005

Let \(K\) be a commutative ring and let \(A\) and \(B\) be \(K\) algebras. The authors define a twisted tensor product of \(A\) and \(B\) to be an algebra \(C\) together with injective algebra homomorphisms \(i_A:A \to C\) and \(i_B:B \to C\) such that the induced linear map \(A\otimes_K B \to C\) is a linear isomorphism.
One way to produce such a \(C\) is from a \(K\) linear map \(\tau:A\otimes B \to B\otimes A\) such that \(\tau(b\otimes 1)=1\otimes b\) and \(\tau(1\otimes a)=a\otimes 1\) and then on \(A\otimes B\) define a multiplication \(\mu_\tau\) by \((\mu_A\otimes \mu_B) \circ (id_A\otimes \tau\otimes id_B)\), where \(\mu_A\), \(\mu_B\) denote the multiplications on \(A\), \(B\). This will be an associative multiplication as long as \(\tau \circ (\mu_B\otimes \mu_A)=\mu_\tau \circ (\tau\otimes \tau) \circ (id_B\otimes \tau\otimes id_A)\), in which case the authors call \(\tau\) a twisting map for \(A\) and \(B\) and write \(A\otimes_\tau B\) for \(A\otimes B\) with multiplication \(\mu_\tau\). They prove that every twisted tensor product arises in this way from a twisting map, and that the map is uniquely determined by the twisted tensor product. They further show that twisting maps extend uniquely to the graded differential algebra of differential forms. They also, for the case \(k=\mathbb{R}\) or \(\mathbb{C}\), construct the analogue of Hochschild cohomology for twisted tensor products: let \(M^n(A\otimes B)=M^n\) denote the \((n + 1)\) linear maps from \(A\otimes B\) to \(A\otimes B\) and let \(M(A\otimes B)=M\) be \(\bigoplus_n M^n\), with the Lie algebra structure such that bracketing with the multiplication of \(A\otimes B\) is the Hochschild differential. They define a subgraded Lie algebra \(C(A\otimes B)\) which for any twisting map \(\tau\) is closed under the differential given by bracketing with \(\tau\). The corresponding cohomology is denoted \(H^*(\tau)\), and they show that it relates in the usual way to the deformations of twisting maps, and provide explicit calculations for two cocycles. These results are applied to deformations of trivial \(\tau\), and hence to deformations of the tensor product itself.
Reviewer: A.R.Magid (Norman)

MSC:

16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S40 Smash products of general Hopf actions
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References:

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