##
**Algebras, bialgebras, quantum groups, and algebraic deformations.**
*(English)*
Zbl 0788.17009

Deformation theory and quantum groups with applications to mathematical physics, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Amherst/MA (USA) 1990, Contemp. Math. 134, 51-92 (1992).

[For the entire collection see Zbl 0755.00012.]

This paper is a review of the past and present state of algebraic deformation theory. It develops along a path which goes from deformation of associative algebras to quantum groups. Let \(k\) be a field and \(k[[t]]\) be its power series ring. A formal deformation of a \(k\)-algebra \(A\) is a \(k[[t]]\)-algebra \(A[[t]]\) endowed with a multiplication \(\mu_ t= \sum_{k=0}^ \infty t^ k \mu_ k\), where \(\mu_ 0(a,b)=ab\) for all \(a\), \(b\) in \(A\) and \(\mu_ k\) \((k\in\mathbb{N}^*)\) is a (\(k\)-bilinear) mapping of \(A^ 2\) into \(A\). The conditions on the multiplication depend on the category in which the deformation takes place. After some generalities about \(k\)-deformations, the authors discuss infinitesimal methods and the relationship of deformations with cohomology. They emphasize several special cases (associative algebras, Lie algebras, …). Every deformation has an infinitesimal which lies in an appropriate second cohomology group and the full cohomology group has a graded Lie product which “controls” the obstructions. The authors define a richer structure that they call a comp algebra, they introduce Schouten brackets, the related \(G\)-algebras [V. Coll, M. Gerstenhaber and S. D. Schack, J. Pure Appl. Algebra 90, No. 3, 201-219 (1993; Zbl 0827.17021)] and Palais pairs [R. Palais, Proc. Symp. Pure Math. 3, 130-137 (1961; Zbl 0126.03404)], and they present universal deformation formulas in the realm of global theory. The paper ends with a review of the deformation theory of presheaves of algebras and with a discussion of bialgebra deformations and cohomology.

This paper is a review of the past and present state of algebraic deformation theory. It develops along a path which goes from deformation of associative algebras to quantum groups. Let \(k\) be a field and \(k[[t]]\) be its power series ring. A formal deformation of a \(k\)-algebra \(A\) is a \(k[[t]]\)-algebra \(A[[t]]\) endowed with a multiplication \(\mu_ t= \sum_{k=0}^ \infty t^ k \mu_ k\), where \(\mu_ 0(a,b)=ab\) for all \(a\), \(b\) in \(A\) and \(\mu_ k\) \((k\in\mathbb{N}^*)\) is a (\(k\)-bilinear) mapping of \(A^ 2\) into \(A\). The conditions on the multiplication depend on the category in which the deformation takes place. After some generalities about \(k\)-deformations, the authors discuss infinitesimal methods and the relationship of deformations with cohomology. They emphasize several special cases (associative algebras, Lie algebras, …). Every deformation has an infinitesimal which lies in an appropriate second cohomology group and the full cohomology group has a graded Lie product which “controls” the obstructions. The authors define a richer structure that they call a comp algebra, they introduce Schouten brackets, the related \(G\)-algebras [V. Coll, M. Gerstenhaber and S. D. Schack, J. Pure Appl. Algebra 90, No. 3, 201-219 (1993; Zbl 0827.17021)] and Palais pairs [R. Palais, Proc. Symp. Pure Math. 3, 130-137 (1961; Zbl 0126.03404)], and they present universal deformation formulas in the realm of global theory. The paper ends with a review of the deformation theory of presheaves of algebras and with a discussion of bialgebra deformations and cohomology.

Reviewer: U.Cattaneo (Maggia)

### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

16S80 | Deformations of associative rings |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |