Symplectic group

Fomenko, A. T. Group symplectic structures on homogeneous spaces. Dokl. Akad. Nauk SSSR 253 (1980), No. 5, 1062-1067. 

Having defined the utility of a waveform library we go on to investigate the utilities of a few libraries. Specifically, we consider libraries generated from a fixed waveform 4>o, usually an unmodulated pulse of some fixed duration, by symplectic transformations. Such transformations form a group of unitary transformations on L2(R) and include linear frequency modulation as well as the Fractional Fourier transform (FrFT) in a sense that we shall make clear. 

It is not hard to see that such transformations form a group. Suppose that Ui and U2 are symplectic in this sense and S and S2 correspond to them. Then... 

In Chapter 5, we have studied Morse theory on a symplectic manifold X given by an action of a compact torus T. As noted there, when X is a Kahler manifold, the gradient flow is given by the associated holomorphic action of the complexification T of T. Hence, the stable and the unstable manifolds can be expressed purely in terms of the group action. 

Let (A, A) be a principally polarized abelian variety over an algebraically closed field k. If the characteristic of k is not equal to the prime p, then the kernel of multiplication by p on A(k) is a finite group isomorphic to (Z/pZ)29. The polarization A induces a nondegenerate alternating pairing A k)[p] x A( )[p] -+ pp(k). Hence, we can try to classify principally polarized abelian varieties with a symplectic basis for the group of points of order p. However, this no longer works in characteristic p. 

R.21+1 (l = 1,2,3), exceptional group G2 and symplectic group Sp4i+2 belong to the above-mentioned groups. 

The relevant Casimir operator of unitary group U21+1 follows from (5.33) if we include in it the term with k = 0. The Casimir operator of symplectic group Sp4i+2, whose generators are tensors Uk and Vk+l with the odd sums of ranks, may be defined in the following way ... 

Combining the above formulas, we can work out analytical expressions for the sums of the scalar products (Tk-Tk) which are generally obtained using the Casimir operators of the unitary group Ify+i and the symplectic group Sp2j+i... 

Maxwell theory, soliton flows are Hamiltonian flows. Such Hamiltonian functions define symplectic structures6 for which there is an absence of local invariants but an infinite-dimensional group of diffeomorphisms which preserve global properties. In the case of solitons, the global properties are those permitting the matching of the nonlinear and dispersive characteristics of the medium through which the wave moves. 

With the connection of PDEs, and especially soliton forms, to group symmetries established, one can conclude that if the Maxwell equation of motion that includes electric and magnetic conductivity is in soliton (SGE) form, the group symmetry of the Maxwell field is SU(2). Furthermore, because solitons define Hamiltonian flows, their energy conservation is due to their symplectic structure. 

A new possibility of supersymmetry arises when n = nv = n. In this case, using fermionic creation and annihilation operators it is possible to construct the generators of the symplectic group Usp (2n). Consequently a supergroup chain starting with decomposition into the orthosymplectic group U (6/2n) Osp (6/2n) (7)... 

Sheaf in fpqc topology 117 Smooth group scheme 88 Solvable group scheme 73 Spec A 41 Split torus 56 Strictly upper triangular 62 Subcomodule 23 Symplectic group 99... 

The symmetric group S(n) is of fundamental importance in quantum chemistry as well in nuclear models and symplectic models of mesoscopic systems. One wishes to discuss the properties of the symmetric group for general n and concentrate on stable results that are essentially n—independent. Here the reduced notation(6)-(9) proves to be very useful. The tensor ir-reps A of S(n) are labelled by ordered partitions(A) of integers where A I- n. In reduced notation the label Ai, A2,. .., Ap for S(n) is replaced by (A2,...,AP). Kronecker products can then be fully developed in a n-independent manner and readily programmed. Thus one finds, for example, the terms arising in the reduced Kronecker product (21) (22) are... 

Thus the symplectic maps form a group under composition. 

This is just the leapfrogA erlet method in its Hamiltonian form. Since we have obtained the method as the composition of two symplectic maps, and the symplectic maps form a group, we know that this method will also be symplectic. 

The characteristic feature of symplectic groups is that they leave invariant antisymmetric bilinear forms [Weyl 1953, ch. VI, eq. (1.1)], a description that applies to the eigenfunction (104). We can regard Sp(14) as an alternative to SOs(3) x U(7)... 

The group structures that have been discussed so far do not exhaust all the possibilities. It has already been pointed out (in section 8.2.1) that U(14) of the chain (52) can have Sp(14) as a subgroup. This symplectic group is identical to the one... 

---