Deformations and the coherence.

*(English)*Zbl 0838.18005
Bureš, J. (ed.) et al., Proceedings of the winter school on geometry and physics, Zdíkov, Czech Republic, January 1993. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 37, 121-151 (1994).

The cotangent cohomology of S. Lichtenbaum and M. Schlessinger [Trans. Am. Math. Soc. 128, 41-70 (1967; Zbl 0156.27201)] is known for its ability to control the deformation of the structure of a commutative algebra.

Considering algebras in the wider sense to include coalgebras, bialgebras and similar algebraic structures such as the Drinfel’d algebras encountered in the theory of quantum groups, one can model such objects as models for an algebraic theory much in the sense of F. W. Lawvere [Proc. Natl. Acad. Sci. USA 50, 869-872 (1963)]. Individual algebras are then determined by their structural constants and hence may be amenable to a deformation theoretic approach for determining their stability under change of these constants.

The basic notion of presentation of an algebraic theory allows the author to develop a deformation cohomology theory, based on the type of construction initially used by Lichtenbaum and Schlessinger, and to start the process of calculation in specific instances. The author notes the connection between the approach used here and the problem of finding coherence proofs in the theory of tensor categories and identifies the relations between the axioms and the relations between the relations as being key problem areas that need more knowledge if effective calculations are to be made in other settings.

For the entire collection see [Zbl 0823.00015].

Considering algebras in the wider sense to include coalgebras, bialgebras and similar algebraic structures such as the Drinfel’d algebras encountered in the theory of quantum groups, one can model such objects as models for an algebraic theory much in the sense of F. W. Lawvere [Proc. Natl. Acad. Sci. USA 50, 869-872 (1963)]. Individual algebras are then determined by their structural constants and hence may be amenable to a deformation theoretic approach for determining their stability under change of these constants.

The basic notion of presentation of an algebraic theory allows the author to develop a deformation cohomology theory, based on the type of construction initially used by Lichtenbaum and Schlessinger, and to start the process of calculation in specific instances. The author notes the connection between the approach used here and the problem of finding coherence proofs in the theory of tensor categories and identifies the relations between the axioms and the relations between the relations as being key problem areas that need more knowledge if effective calculations are to be made in other settings.

For the entire collection see [Zbl 0823.00015].

Reviewer: T.Porter (Bangor)

##### MSC:

18G60 | Other (co)homology theories (MSC2010) |

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

18C10 | Theories (e.g., algebraic theories), structure, and semantics |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |