Publication: El grupo simpléctico real
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2017-06
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En este trabajo se estudia el grupo simpléctico real de los endomorfismos lineales que preservan una forma bilineal antisimétrica no degenerada. Un acercamiento algebraico básico muestra que coincide con su subgrupo especial y permite obtener generadores, las transvecciones simplécticas. Desde un punto de vista geométrico, se ve que el grupo simpléctico es un grupo de Lie y se estudia su algebra de Lie asociada. Su subgrupo ortogonal es un subgrupo de Lie compacto isomorfo al grupo unitario y, además, es un retracto de deformación del grupo simpléctico. Para concluir todo esto se desarrollan herramientas analíticas como la aplicación exponencial, el logaritmo y las potencias de reales de matrices simétricas definidas positivas.
In this work we study the real symplectic group consisting of linear endomorphisms that preserve an antisymmetric bilinear form. A basic algebraic approach shows that its special group is the whole group and provides generators known as sym plectic transvections. From the geometric viewpoint we see that the symplectic group is a Lie group to which a Lie algebra can be associated. Its orthogonal subgroup is a compact Lie subgroup isomorphic to the unitary group and, furthermore, it is a deformation retract of the symplectic group. Analytical tools such as the exponential mapping, the logarithm and real powers of definite positive symmetric matrices are developed in the process.
In this work we study the real symplectic group consisting of linear endomorphisms that preserve an antisymmetric bilinear form. A basic algebraic approach shows that its special group is the whole group and provides generators known as sym plectic transvections. From the geometric viewpoint we see that the symplectic group is a Lie group to which a Lie algebra can be associated. Its orthogonal subgroup is a compact Lie subgroup isomorphic to the unitary group and, furthermore, it is a deformation retract of the symplectic group. Analytical tools such as the exponential mapping, the logarithm and real powers of definite positive symmetric matrices are developed in the process.
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