Crossed products by a coalgebra

We introduce the notion of a crossed product of an al¬gebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.

[1]  S. Majid,et al.  Coalgebra Gauge Theory , 1996 .

[2]  Doi Yukio,et al.  Cleft comodule algebras for a bialgebra , 1986 .

[3]  K. Schmüdgen.  Integrable operator representations of ℝq2,Xq,γ andSLq(2, ℝ) , 1994 .

[4]  M. Sweedler.  Cohomology of algebras over Hopf algebras , 1968 .

[5]  H. Schneider.  Normal basis and transitivity of crossed products for Hopf algebras , 1992 .

[6]  S. Majid.  Braided groups and algebraic quantum field theories , 1991 .

[7]  S. Majid.  Foundations of Quantum Group Theory , 1995 .

[8]  S. Majid.  Beyond Supersymmetry and Quantum Symmetry (an introduction to braided groups and braided matrices) , 1992, hep-th/9212151.

[9]  Crossed Product Structure of Quantum Euclidean Groups , 1996, q-alg/9612013.

[10]  Quantum groups on fibre bundles , 1994, hep-th/9401085.

[11]  Shahn Majid,et al.  Cross Products by Braided Groups and Bosonization , 1994 .

[12]  S. Woronowicz.  Unbounded elements affiliated withC*-algebras and non-compact quantum groups , 1991 .

[13]  Yuri Bespalov,et al.  Hopf (bi-)modules and crossed modules in braided monoidal categories , 1995 .

[14]  Yukio Doi.  Equivalent crossed products for a hopf algebra , 1989 .

[15]  Susan Montgomery,et al.  Crossed products and inner actions of Hopf algebras , 1986 .

[16]  F. Bonechi,et al.  Freeq-Schrödinger equation from homogeneous spaces of the 2-dim Euclidean quantum group , 1996 .

[17]  S. Montgomery,et al.  Crossed products and Galois extensions of Hopf algebras. , 1989 .

[18]  F.Bonechi,et al.  Free q-Schrodinger Equation from Homogeneous Spaces of the 2-dim Euclidean Quantum Group , 1994, hep-th/9409048.