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A complex symplectic structure on a Lie algebra h is an integrable complex structure J with a closed non-degenerate (2,0)-form. It is determined by J and the real part Ω of the (2,0)-form. Suppose that h is a semi-direct product g?V, and both g and V are Lagrangian with respect to Ω and totally real with respect to J. This note shows that g?V is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω and J are isomorphic. 相似文献
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A new Poisson structure is defined on a subspace of the Kupershmidt algebra, isomorphic to the space H of n×n Hermitian matrices. The new Poisson structure is of Lie–Poisson type with respect to the standard Lie bracket of H. This Poisson structure (together with two already known ones, obtained through a r-matrix technique) allows to construct an extension of the periodic Toda lattice with n particles that fits in a trihamiltonian recurrence scheme. Some explicit examples of the construction and of the first integrals found in this way are given. 相似文献
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We construct a natural L2-metric on the perturbed Seiberg–Witten moduli spaces Mμ+ of a compact 4-manifold M, and we study the resulting Riemannian geometry of Mμ+. We derive a formula which expresses the sectional curvature of Mμ+ in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U(1) bundle P→Mμ+ such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M, the L2-metric on Mμ+ coincides with the natural Kähler metric on moduli spaces of vortices. 相似文献
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We introduce the notion of omni-Lie 2-algebra, which is a categorification of Weinstein’s omni-Lie algebras. We prove that there is a one-to-one correspondence between strict Lie 2-algebra structures on 2-sub-vector spaces of a 2-vector space V and Dirac structures on the omni-Lie 2-algebra gl(V)⊕V. In particular, strict Lie 2-algebra structures on V itself one-to-one correspond to Dirac structures of the form of graphs. Finally, we introduce the notion of twisted omni-Lie 2-algebra to describe (non-strict) Lie 2-algebra structures. Dirac structures of a twisted omni-Lie 2-algebra correspond to certain (non-strict) Lie 2-algebra structures, which include string Lie 2-algebra structures. 相似文献
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We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Zp) gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n dimensional vector space which we call Hn. The Zp gauge particles act on the vertex particles and thus Hn can be thought of as a C(Zp) module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n and p, though we believe this feature holds for all n>p. We will see that non-Abelian anyons of the quantum double of C(S3) are obtained as part of the vertex excitations of the model with n=6 and p=3. Ising anyons are obtained in the model with n=4 and p=2. The n=3 and p=2 case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Zp. This makes them possible candidates for realizing quantum computation. 相似文献
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We consider a complete nonnegative biminimal submanifold M (that is, a complete biminimal submanifold with λ≥0) in a Euclidean space EN. Assume that the immersion is proper , that is, the preimage of every compact set in EN is also compact in M. Then, we prove that M is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0, we construct examples of biminimal submanifolds and curves. 相似文献
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A curve α immersed in the three-dimensional sphere S3 is said to be a Bertrand curve if there exists another curve β and a one-to-one correspondence between α and β such that both curves have common principal normal geodesics at corresponding points. The curves α and β are said to be a pair of Bertrand curves in S3. One of our main results is a sort of theorem for Bertrand curves in S3 which formally agrees with the classical one: “Bertrand curves in S3 correspond to curves for which there exist two constants λ≠0 and μ such that λκ+μτ=1”, where κ and τ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S3 as the only twisted curves in S3 having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S3 and (1,3)-Bertrand curves in R4. 相似文献
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In this paper we show that all supergravity billiards corresponding to σ -models on any U/H non-compact-symmetric space and obtained by compactifying supergravity to D=3 admit a closed form general integral depending analytically on a complete set of integration constants. The key point in establishing the integration algorithm is provided by an upper triangular embedding of the solvable Lie algebra associated with U/H into sl(N,R) which is guaranteed to exist for all non-compact symmetric spaces and also for homogeneous special geometries non-corresponding to symmetric spaces. In this context we establish a remarkable relation between the end-points of the time-flow and the properties of the Weyl group. The asymptotic states of the developing Universe are in one-to-one correspondence with the elements of the Weyl group which is a property of the Tits–Satake universality classes and not of their single representatives. Furthermore the Weyl group admits a natural ordering in terms of ?T, the number of reflections with respect to the simple roots. The direction of time flows is always from the minimal accessible value of ?T to the maximum one or vice versa. 相似文献