Broken or deformed symmetry is constructed as a multiparametric family of groups (or algebras) with boundaries containing the initial (nondeformed) ones.

The application of group (and algebraic) deformation theory in supergravity and gauge models of elementary particles interactions provides the possibility to obtain important results. For example, the boundary model "remember" the global invariants of the deformed symmetry and obeys the restrictions that cannot be described only in terms of the limit symmetry. (See these and other examples in the publications [1-4]).

Quantizations of Poisson systems that are "living" on group manifolds and homogenious spaces are studied as violations (deformations) of the primitive (commutative) Poisson-Lie algebra in the "direction" of some admissable Poisson bracket. A group or a homogenious space is considered as a Hopf algebra. The quantization is its deformation. This is why the deformed Hopf algebras and in particular quantum groups and algebras are so popular now in quantum field theory. (See, for example, the papers [5-10].)

Composing the deformed symmetry and investigating its properties one must use the differential geometry and algebraic topology. To solve the problems arising in the process of deformation quantization the group theoretical instruments and new algebraic methods must be developed. (See for example the papers [11-15].)

2. Lyakhovsky V.D., Vasilievich D.V., Broken CP-invariance in d=11 Supergravitation, Nuclear Physics, 43, N 5, p.p. 1296-1297, 1986

3. Lyakhovsky V.D., Shtykov N.N., The Mass Spectrum in N=11 Supergravitation with SU(3)*U(1)/U(1)*U(1)-Compactification, Nuclear Physics, 46, N 1, p.p. 273-280, 1987

4. Lyakhovsky V.D., Vasilievich D.V., Algebraic Approach to Kaluza-Klein Models, Letters in Mathematical Physics, 17, p.p. 109-115, 1989

5. Lyakhovsky V.D., Mirolyubov A.M., Contractions in Deformed Lie-Poisson Structures, Intern. Journ. Modern Phys. A, 12, N 1, p.p. 225-230, 1997

6. Lyakhovsky V.D., Tkach V.I., BRST Quantum Algebra, its Double and R-matrix, Journal of Physics, A: Math. Gen., 31, p.p. 2869-2880, 1998

7. Kulish P.P., Lyakhovsky V.D., Mudrov A.I., Extended Jordanian twists for Lie algebras, Journal of Mathematical Physics, 40, p.p. 2632-2645, 1999

8. Kulish P.P., Lyakhovsky V.D., Jordanian twists on deformed carrier spaces, Journal of Physics, A: Math. Gen., 33, p.p. L279-L285, 2000

9. Kulish P.P., Lyakhovsky V.D., Stolin A.A., Chains of Frobenius subalgebras of so(M) and the corresponding twists, Journal of Mathematical Physics, 42, p.p. 5006-5019, 2001

10. Lukierski J., Mozrzymas M., Lyakhovsky V., \kappa-Deformations of D=4 Weil and Conformal Symmetries, Physics Letters B, 538, p.p.375-389, 2002

11. Lyakhovsky V.D., Algebraic Constructions of Spin Structures on Homogeneous Spaces, Kluwer Academic, in "Spinors, Twistors, Clifford Algebras and Quantum Deformations", Netherlands, p.p. 31-38, 1993

12. Lyakhovsky V.D., Melnikov S.Yu., Recursion Relations and Branching Rules for Simple Lie Algebras, Journal of Physics, A: Math. Gen., 29, p.p. 1075-1087, 1996

13. Lyakhovsky V.D., Quantum Duality. Classical and Quasiclassical Limits, Czechoslovak Journal of Physics, 46, N 2/3, p.p. 227-234, 1996

14. Lyakhovsky V.D., Regular and Special Subgroups, Saint-Petersburg University Publishing Company, 30p., 1999

15. Kwek L.C., Lyakhovsky V.D., Cohomologies and Peripheric chains, Czech. Journal of Physics, 51, p.p. 1374-1379, 2001