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Rather, it is isomorphic to a subgroup of Sp(2n, C), and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of Sp(n) is the compact real form of the complex symplectic Lie algebra sp(2n, C). Sp(n) is a real Lie group with (real) dimension n(2n + 1). It is compact and simply ...
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Since SU(n) is simply connected, [2] we conclude that SL(n, C) is also simply connected, for all n greater than or equal to 2. The topology of SL(n, R) is the product of the topology of SO(n) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as ...
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The real numbers are more numerous than the natural numbers. Moreover, R {\displaystyle \mathbb {R} } has the same number of elements as the power set of N {\displaystyle \mathbb {N} } . Symbolically, if the cardinality of N {\displaystyle \mathbb {N} } is denoted as ℵ 0 {\displaystyle \aleph _{0}} , the cardinality of the continuum is
The general linear group over the field of complex numbers, GL(n, C), is a complex Lie group of complex dimension n 2. As a real Lie group (through realification) it has dimension 2n 2. The set of all real matrices forms a real Lie subgroup. These correspond to the inclusions GL(n, R) < GL(n, C) < GL(2n, R),
The Schur multiplier of the discrete group PSL(2, R) is much larger than Z, and the universal central extension is much larger than the universal covering group. However these large central extensions do not take the topology into account and are somewhat pathological.
In particular it satisfies s N for N > n and therefore it is a PI-ring. If R and S are PI-rings then their tensor product over the integers, , is also a PI-ring. If R is a PI-ring, then so is the ring of n × n matrices with coefficients in R.