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Yang-Baxter operators and noncommutative de Rham complexes. (English. Russian original) Zbl 0846.17016
Russ. Acad. Sci., Izv., Math. 44, No. 2, 315-338 (1995); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 58, No. 2, 108-131 (1994).
In the first part the author introduces several notions following Manin’s approach to quantum groups and spaces. A quantum matrix semigroup (over a field $${\mathbf k}$$) is a bialgebra $$M={\mathbf k}\langle Z\rangle/ (I_M)$$ generated by entries of an $$n\times n$$ matrix $$Z$$, with the usual comultiplication and counit. It is called a weak $$R$$-matrix semigroup if $$I_M= RZ\circ Z- Z\circ ZR$$, with $$R:{\mathbf k}^{n^2}\to {\mathbf k}^{n^2}$$ and $$(Z\circ Z)^{kl}_{ij}= z^k_i z^l_j$$. This is a strong $$R$$-matrix semigroup if $$R$$ verifies the Yang-Baxter equation. One relates to $$M$$ a matrix algebra $$R_{\text{alg}} (M)$$ formed by matrices $$S$$ such that $$SZ\circ Z- Z\circ ZS=0$$ in $$M$$. A quantum space $$A$$ is a (left) comodule over $$M$$ and to an $$s$$-tuple $$A_1, \dots, A_s$$ one relates a universal quantum semigroup $$M(A_1, \dots, A_s)$$. Mutual relations between all these notions are studied in detail.
These results are applied in the second part of the paper where noncommutative de Rham complexes are defined axiomatically. For a couple of $$n^2 \times n^2$$ matrices $${\mathcal A}$$ and $${\mathcal B}$$ and for two $$n$$-vectors of generators $$x$$ and $$\xi$$, with $$\xi_i= dx_i$$, one sets $$A= {\mathbf k}\langle x\rangle/ (x\circ x- {\mathcal A}x\circ x)$$, $$B= {\mathbf k}\langle \xi\rangle/( \xi\circ \xi- {\mathcal B}\xi\circ \xi)$$. De Rham complex $$\Lambda (A, B)$$ contains $$A$$ and $$B$$ as subalgebras and $$M(A, B)$$ coacts on $$\Lambda (A, B)$$. The conditions on existence and the classification are described. The final part is devoted to differential calculus on quantum semigroups and, first of all, to working out some basic examples.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 58A12 de Rham theory in global analysis 16W25 Derivations, actions of Lie algebras 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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