Cross products by braided groups and bosonization.

*(English)*Zbl 0807.16036The author continues the study of certain algebraic structures living in braided monoidal (or quasitensor) categories. [See also the author’s survey paper, in Advances in Hopf algebras, 55-105 (1994; Zbl 0812.18004).]

Let \(H\) be a cocommutative Hopf algebra and \(B\) a Hopf algebra which is a left \(H\)-module bialgebra. For these data a cross product Hopf algebra \(B\rtimes H\) can be defined (Molnar, 1977). The author obtains two generalizations of this result:

1) a braided variant where \(H\) is a braided cocommutative Hopf algebra (braided group) and \(B\) is a (quasitriangular) Hopf algebra and both live in an arbitrary quasitensor category;

2) a bosonization construction which turns any Hopf algebra \(B\) in the braided category \(_H{\mathcal M}\) of modules over a quasitriangular Hopf algebra \(H\) into the ordinary Hopf algebra \(\text{bos}(B)=B\rtimes H\). (The origin of this term stems from physics where \(\mathbb{Z}_2\)-graded algebras etc. are called ‘fermionic’ while ordinary ungraded ones are called ‘bosonic’.)

Earlier the author defined in some sense an adjoint procedure of transmutation which turns an ordinary Hopf algebra with quasitriangular Hopf algebra maps into a Hopf algebra in the category \(_H{\mathcal M}\). This allows the author to obtain the bosonization theorem from the braided variant of Molnar’s theorem.

Using the notion of quantum braided group (or quasitriangular Hopf algebra in a braided category), introduced by the author, it is possible to consider both these theorems as special cases of a braided variant of the bosonization theorem [cf. the reviewer, “Crossed modules and quantum groups in braided categories I, II”, preprints ITP-94-21E, ITP-94-38E, Kiev].

Let \(H\) be a cocommutative Hopf algebra and \(B\) a Hopf algebra which is a left \(H\)-module bialgebra. For these data a cross product Hopf algebra \(B\rtimes H\) can be defined (Molnar, 1977). The author obtains two generalizations of this result:

1) a braided variant where \(H\) is a braided cocommutative Hopf algebra (braided group) and \(B\) is a (quasitriangular) Hopf algebra and both live in an arbitrary quasitensor category;

2) a bosonization construction which turns any Hopf algebra \(B\) in the braided category \(_H{\mathcal M}\) of modules over a quasitriangular Hopf algebra \(H\) into the ordinary Hopf algebra \(\text{bos}(B)=B\rtimes H\). (The origin of this term stems from physics where \(\mathbb{Z}_2\)-graded algebras etc. are called ‘fermionic’ while ordinary ungraded ones are called ‘bosonic’.)

Earlier the author defined in some sense an adjoint procedure of transmutation which turns an ordinary Hopf algebra with quasitriangular Hopf algebra maps into a Hopf algebra in the category \(_H{\mathcal M}\). This allows the author to obtain the bosonization theorem from the braided variant of Molnar’s theorem.

Using the notion of quantum braided group (or quasitriangular Hopf algebra in a braided category), introduced by the author, it is possible to consider both these theorems as special cases of a braided variant of the bosonization theorem [cf. the reviewer, “Crossed modules and quantum groups in braided categories I, II”, preprints ITP-94-21E, ITP-94-38E, Kiev].

Reviewer: Yu.Bespalov (Kiev)

##### MSC:

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

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

17A70 | Superalgebras |

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

17B81 | Applications of Lie (super)algebras to physics, etc. |