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In this research, we studied the affine Lie algebra aff(2,R). The aim of this research is to determine the 1-form in affine Lie algebra aff(2,R) which is associated with its symplectic structure so that affine Lie algebra aff(2,R) is a Frobenius Lie algebra. Realized the elements of the affine Lie algebra aff(2,R) in matrix form, then calculated the Lie brackets and formed the structure matrix of the affine Lie algebra aff(2,R). 1-form of the affine Lie algebra aff(2,R) is obtained from the determinant of the structure matrix of the affine Lie algebra aff(2,R). Furthermore, proved that the 2-form is symplectic and related to the 1-form. The result obtained is that the affine Lie algebra aff(2,R) has 1-form α=ε_12^*+ε_23^* on aff(2,R)^* which is related to its symplectic structure, β=ε_11^*∧ε_12^*+ε_12^*∧ε_22^*+ε_21^*∧ε_13^*+ε_22^*∧ε_23^* such that the affine Lie algebra aff(2,R) is a Frobenius Lie algebra. For further research, it can be developed into an affine Lie algebra with dimensions n(n+1).

In this work, we have characterized the frame bundle F M admitting metallic structures on almost quadratic ϕ -manifolds ϕ 2 = p ϕ + q I − q η ⊗ ζ , where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ -structure to the metallic structure on F M are constructed. We also prove the existence of a metallic structure on F M with the aid of the J ˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J ˜ on F M . Finally, we construct an example of it to finish. [ABSTRACT FROM AUTHOR]

In this work, we have characterized the frame bundle FM admitting metallic structures on almost quadratic ϕ-manifolds ϕ2=pϕ+qI−qη⊗ζ, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic ϕ-structure to the metallic structure on FM are constructed. We also prove the existence of a metallic structure on FM with the aid of the J˜ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field J˜ on FM. Finally, we construct an example of it to finish.

The probability distribution of the magnitude C of the curvature 2-form, that underlies the quantum geometric phase and the reaction force of geometric magnetism, is calculated for an ensemble of three-parameter Hamiltonians represented by the gaussian unitary ensemble of N × N matrices. The distributions are determined analytically: exactly for N = 2 and approximately for N ≥ 3, and compared with simulations. The distributions decay asymptotically as 1/C5/2; this is a consequence of the codimension of energy-level degeneracies in the ensemble. [ABSTRACT FROM AUTHOR]

In this work, over n-dimensional (n > 4) oriented inner product space V, duality operator TΦ is defined by using the non zero ,(n-4)-form Φ and it is shown that this operator is symmetric. By the means of operator TΦ , Over the spaces whose dimension is greater than four we defined self-duality, anti-self-duality, weak self-duality and weak anti-self-duality of a 2-form. Especially, over the spaces Rn for 5<=n<=8 the duality operator TΦ which corresponds to the fundamental forms is studied in details. [ABSTRACT FROM AUTHOR]