arXiv subjects

Ranee Brylinski

Publications and source records attributed to Ranee Brylinski.

12 recordsLinked to original sources

arXiv2003

Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models I

We attach a Dixmier algebra B to the closure of any nilpotent orbit of G where G is GL(n,C), O(n,C) or Sp(2n,C). This algebra B is a noncommutative analog of the coordinate ring R of the orbit closure, in the sense that B has a G-invariant algebra filtration and gr B=R. We obtain B by making a noncommutative analog of the Kraft-Procesi construction which modeled the orbit closure as the algebraic symplectic reduction of a finite-dimensional symplectic vector space L. Indeed B is a subquotient of the Weyl algebra for L.

math.RT↗

arXiv2003

Complexity and Completeness of Immanants

Immanants are polynomial functions of n by n matrices attached to irreducible characters of the symmetric group S_n, or equivalently to Young diagrams of size n. Immanants include determinants and permanents as extreme cases. Valiant proved that computation of permanents is a complete problem in his algebraic model of NP theory, i.e., it is VNP-complete. We prove that computation of immanants is VNP-complete if the immanants are attached to a family of diagrams whose separation is $Ω(n^δ)$ for some $δ>0$. We define the separation of a diagram to be the largest number of overhanging boxes contained in a single row. Our theorem proves a conjecture of Buergisser for a large variety of families, and in particular we recover with new proofs his VNP-completeness results for hooks and rectangles.

cs.CC↗

arXiv2001

Universal quantum gates

In this paper we study universality for quantum gates acting on qudits.Qudits are states in a Hilbert space of dimension d where d is at least two. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V, and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors.

quant-ph↗

arXiv2001

Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

We construct a unique G-equivariant graded star product on the algebra $S(g)/I$ of polynomial functions on the minimal nilpotent coadjoint orbit $\Omin$ of G where G is a complex simple Lie group and $g\neq\sl_2(C)$. This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to compute, for G classical, the star product of a momentum function $μ_x$ with any function f. We find $μ_x\star f=μ_xf+\half\{μ_x,f\}t+Λ^x(f)t^2$. For $\g$ different from $sp_n(\C)$, $Λ^x$ is not a differential operator. Instead $\Lamda^x$ is the left quotient of an explicit order 4 algebraic differential operator $D^x$ by an order 2 invertible diagonalizable operator. Precisely, $Λ^x=-{1/4}\frac{1}{E'(E'+1)}D^x$ where $E'$ is a positive shift of the Euler vector field. Thus $μ_x\star f$ is not local in f. Using $\star$ we construct a positive definite hermitian inner product on $Sg/I$. The Hilbert space completion of $Sg/I$ is then a unitary representation of $G$. This quantizes $\Omin$ in the sense of geometric quantization and the orbit method.

math.QA↗

arXiv2000

Invariant Polynomial Functions on k qudits

We study the polynomial functions on tensor states in $(C^n)^{\otimes k}$ which are invariant under $SU(n)^k$. We describe the space of invariant polynomials in terms of symmetric group representations. For $k$ even, the smallest degree for invariant polynomials is $n$ and in degree $n$ we find a natural generalization of the determinant. For $n,d$ fixed, we describe the asymptotic behavior of the dimension of the space of invariants as $k\to\infty$. We study in detail the space of homogeneous degree 4 invariant polynomial functions on $(C^2)^{\otimes k}$.

quant-ph↗

arXiv2000

Equivariant Deformation Quantization for the Cotangent Bundle of a Flag Manifold

Let $\XR$ be a (generalized) flag manifold of a non-compact real semisimple Lie group $\GR$, where $\XR$ and $\GR$ have complexifications X and G. We investigate the problem of constructing a graded star product on $Pol(T^*\XR)$ which corresponds to a $\GR$-equivariant quantization of symbols into smooth differential operators acting on half-densities on $\XR$. We show that any solution is algebraic in that it restricts to a G-equivariant graded star product star on the algebraic part R of $Pol(T^*\XR)$. We construct, when R is generated by the momentum functions $μ^x$ for G, a preferred choice of star where $μ^x\starϕ$ has the form $μ^xϕ+\half\{μ^x,ϕ\}t+Λ^x(ϕ)t^2$. Here $Λ^x$ are operators on R which are not differential in the known examples and so $μ^x\starϕ$ is not local in $ϕ$. R acquires an invariant positive definite inner product compatible with its grading. The completion of R is a new Fock space type model of the unitary representation of G on $L^2$ half-densities on X.

math.QA↗

arXiv2000

Non-Locality of Equivariant Star Products on $T*(RP^n)$

Lecomte and Ovsienko constructed $SL_{n+1}(R)$-equivariant quantization maps $Q_λ$ for symbols of differential operators on $λ$-densities on $\RP^n$. We derive some formulas for the associated graded equivariant star products $star_λ$ on the symbol algebra $Pol(T*\RP^n)$. These give some measure of the failure of locality. Our main result expresses (for $n$ odd) the coefficients $C_p$ of $star_λ$ when $λ=\half$ in terms of some new $SL_{n+1}(C)$-invariant algebraic bidifferential operators $Z_p$ on $T*\CP^n$ and the operators $(E+\frac{n}{2}\pm s)^{-1}$ where $E$ is the fiberwise Euler vector field and $s\in\{1,2,...,[\frac{p}{2}]\}$.

math.QA↗

arXiv2000

Quantization of Double Covers of Nilpotent Coadjoint Orbits I: Noncommutative Models

We construct by geometric methods a noncommutative model E of the algebra of regular functions on the universal (2-fold) cover M of certain nilpotent coadjoint orbits O for a complex simple Lie algebra g. Here O is the dense orbit in the cotangent bundle of the generalized flag variety X associated to a complexified Cartan decomposition g=(p^+)+k+(p^-) where p^+- are Jordan algebras by the TKK construction. We obtain E as the algebra of g-finite differential operators on a smooth Lagrangian subvariety in M where g is given by differential operators twisted according to a critical parameter. After Fourier transform, E is a quadratic extension of the algebra of twisted differential operators for a (formal) tensor power of the canonical bundle. Not only is E a Dixmier algebra for M, in the sense of the orbit method, but also E has a lot of additional structure,including an anti-automorphism, a supertrace, and a non-degenerate supersymmetric bilinear pairing. We show that E is the specialization at t=1 of a graded (non-local) equivariant star product with parity.

math.QA↗

arXiv2000

From Dixmier algebras to Star Products

Let M be a Galois cover of a nilpotent coadjoint orbit of a complex semisimple Lie group. We define the notion of a PERFECT Dixmier algebra for M and show how this produces a graded (non-local) equivariant star product on M with several very nice properties. This is part of a larger program we have been developing for working out the orbit method for nilpotent orbits.

math.QA↗

arXiv1998

Instantons and Kaehler Geometry of Nilpotent Orbits

The first obstacle in building a Geometric Quantization theory for nilpotent orbits of a real semisimple Lie group has been the lack of an invariant polarization. In order to generalize the Fock space construction of the quantum mechanical oscillator, a polarization of the symplectic orbit invariant under the maximal compact subgroup is required. In this paper, we explain how such a polarization on the orbit arises naturally from the work of Kronheimer and Vergne. This occurs in the context of hyperkaehler geometry. The polarization is complex and in fact makes the orbit into a (positive) Kaehler manifold. We study the geometry of this Kaehler structure, the Vergne diffeomorphism, and the Hamiltonian functions giving the symmetry. We indicate how all this fits into a quantization program.

math.SG↗

arXiv1998

Geometric Quantization of Real Minimal Nilpotent Orbits

In this paper, we begin a quantization program for nilpotent orbits of a real semisimple Lie group. These orbits and their covers generalize the symplectic vector space. A complex structure polarizing the orbit and invariant under a maximal compact subgroup is provided by the Kronheimer-Vergne Kaehler structure. We outline a geometric program for quantizing the orbit with respect to this polarization. We work out this program in detail for minimal nilpotent orbits in the non-Hermitian case. The Hilbert space of quantization consists of holomorphic half-forms on the orbit. We construct the reproducing kernel. The Lie algebra acts by explicit pseudo-differential operators on half-forms where the energy operator quantizing the Hamiltonian is inverted. The Lie algebra representation exponentiates to give a minimal unitary ladder representation. Jordan algebras play a key role in the geometry and the quantization.

math.SG↗

arXiv1992

Nilpotent orbits, normality, and Hamiltonian group actions

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the unique maximal semisimple Lie subalgebra of $R=R(M)$ containing $\g$. The action of $\g'\simeq R[2]$ exponentiates to an action of the corresponding Lie group $G'$ on a $G'$-cover $M'$ of a nilpotent orbit in $\g'$ such that $M$ is open dense in $M'$. We determine all such pairs $(\g\subset\g')$.

math.RT↗