Dual Bialgebra Research Articles

Exotic bialgebras from 9 × 9 unitary braid matrices

The exotic bialgebras that arise from a 9 × 9 unitary braid matrix are constructed. The dual bialgebra of one of these exotic bialgebras is presented.

Exotic bialgebras from 9×9 braid matrices

We construct exotic bialgebras that arise from multiparameter 9 × 9 R-matrices, some of which are new. We also construct the dual bialgebras of two of these exotic bialgebras.

Duality and representations for new exotic bialgebras

We find the exotic matrix bialgebras which correspond to the two nontriangular nonsingular 4×4 R-matrices in the classification of Hietarinta, namely, RS0,3 and RS1,4. We find two new exotic bialgebras S03 and S14 which are not deformations of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of RS1,4 in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GLp,q(2).

A Pairing Theorem between a Braided Bialgebra and Its Dual Bialgebra

This paper constructs a dual bialgebra for the FRT-bialgebra related to a Hayashi's one parameter R-matrix by using the skew-derivation method. The natural pairing between them is non-degenerate.

A pairing theorem between multi-parameter bialgebra and its dual bialgebra

This paper constructs a dual bialgebra for a multi-parameter FRT-bialgebra related to a Hayashi’sR-matrix by using the skew-derivation method and proves that the pairing between them is non-degenerate.

On a Hopf Algebra in Graph Theory

We introduce and start the study of a bialgebra of graphs, which we call the 4- bialgebra , and of the dual bialgebra of 4- invariants . The 4-bialgebra is similar to the ring of graphs introduced by W. T. Tutte in 1946, but its structure is more complicated. The roots of the definition are in low dimensional topology, namely, in the recent theory of Vassiliev knot invariants. In particular, 4-invariants of graphs determine Vassiliev invariants of knots. The relation between the two notions is discussed.

Braided antisymmetrizer as bialgebra homomorphism

For a Yang-Baxter operator we show that a bialgebra homomorphism from a free braided tensor bialgebra to a cofree braided shuffle bialgebra is the Woronowicz braided antisymmetrizer. A cofree braided shuffle bialgebra is a braided generalization of a cofree shuffle bialgebra introduced by Sweedler. Its graded dual bialgebra is a free braided tensor bialgebra.

On the continuous dual of a polynomial bialgebra

This paper is devoted to the study of the dual bialgebra of the bialgebra B n of polynomials in finitely many variables. The elements of are described in various ways in orcer to obtain some useful formulae (e.g., a formula for the “remainder” modulo a cofinite ideal of B n). The subcoalgebra C f generated by a given element of is studied and its structure constants with respect to different natural bases are given.