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Register a new userSpecial symplectic Lie groups and hypersymplectic Lie groups
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A special symplectic Lie group is a triple $(G,omega, abla)$ such that $G$ is a finite-dimensional real Lie group and $omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure $ abla$. In this paper starting from a special symplectic Lie group we show how to ``deform the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure $ abla$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.
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The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space $H$ is given by the third cohomology $text{H}^3(H, Bbb Z)$. When $H$ is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of $H$. We shall study in more detail this relation in the case of a group extension $1to N to G to H to 1$ when the gerbe is defined by an abelian extension $1to A to hat N to N to 1$ of $N$. In particular, when $text{H}_s^1(N,A)$ vanishes we shall construct a transgression map $text{H}^2_s(N, A) to text{H}^3_s(H, A^N)$, where $A^N$ is the subgroup of $N$-invariants in $A$ and the subscript $s$ denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. The obtained not Laurent-type polynomials are proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretization. Recurrence relations are shown for the Lie groups of types $A_1$, $A_2$, $A_3$, $C_2$, $C_3$, $G_2$, and $B_3$ together with lowest polynomials.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
Let K be a connected compact semisimple Lie group and Kc its complexification. The generalized Segal-Bargmann space for Kc, is a space of square-integrable holomorphic functions on Kc, with respect to a K-invariant heat kernel measure. This space is connected to the Schrodinger Hilbert space L^2(K) by a unitary map, the generalized Segal-Bargmann transform. This paper considers certain natural operators on L^2(K), namely multiplication operators and differential operators, conjugated by the generalized Segal-Bargmann transform. The main results show that the resulting operators on the generalized Segal-Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on Kc. I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.
We consider random stochastic matrices $M$ with elements given by $M_{ij}=|U_{ij}|^2$, with $U$ being uniformly distributed on one of the classical compact Lie groups or associated symmetric spaces. We observe numerically that, for large dimensions, the spectral statistics of $M$, discarding the Perron-Frobenius eigenvalue $1$, are similar to those of the Gaussian Orthogonal ensemble for symmetric matrices and to those of the real Ginibre ensemble for non-symmetric matrices. Using Weingarten functions, we compute some spectral statistics that corroborate this universality. We also establish connections with some difficult enumerative problems involving permutations.
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