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46
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 43 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
CENTAUR: The system
 In Software Development Environments (SDE
, 1988
"... asymptotic normality for finite dimensional quantum ..."
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Cited by 43 (0 self)
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asymptotic normality for finite dimensional quantum
On quaternions and octonions: their geometry, arithmetic, and symmetry, A K
, 2003
"... Conway and Smith’s book is a wonderful introduction to the normed division algebras: the real numbers (R), the complex numbers (C), the quaternions (H), and the octonions (O). The first two are wellknown to every mathematician. In contrast, the quaternions and especially the octonions are sadly neg ..."
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Cited by 37 (0 self)
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Conway and Smith’s book is a wonderful introduction to the normed division algebras: the real numbers (R), the complex numbers (C), the quaternions (H), and the octonions (O). The first two are wellknown to every mathematician. In contrast, the quaternions and especially the octonions are sadly neglected, so the authors rightly concentrate on these. They develop these number systems from scratch, explore their connections to geometry, and even study number theory in quaternionic and octonionic versions of the integers. Conway and Smith warm up by studying two famous subrings of C: the Gaussian integers and Eisenstein integers. The Gaussian integers are the complex numbers x + iy for which x and y are integers. They form a square lattice:
A Classification of Multiplicity Free Representations
, 1998
"... Let G be a connected reductive linear algebraic group over C and let (ρ, V) be a regular representation of G. There is a locally finite representation (ˆρ, C[V]) on the affine algebra C[V] of V defined by ˆρ(g)f(v) = f(g−1v) for f ∈ C[V]. Since G is reductive, (ˆρ, C[V]) decomposes as a direct sum ..."
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Cited by 24 (0 self)
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Let G be a connected reductive linear algebraic group over C and let (ρ, V) be a regular representation of G. There is a locally finite representation (ˆρ, C[V]) on the affine algebra C[V] of V defined by ˆρ(g)f(v) = f(g−1v) for f ∈ C[V]. Since G is reductive, (ˆρ, C[V]) decomposes as a direct sum of irreducible regular representations of G. The representation (ρ, V) is said to be multiplicity free if each irreducible representation of G occurs at most once in (ˆρ, C[V]). Kac has classified all irreducible multiplicity free representations. In this paper, we classify arbitrary regular multiplicity free representations, and for each new multiplicity free representation we determine the monoid of highest weights occurring in its affine algebra.
Standard Monomial Theory for BottSamelson Varieties
, 1997
"... We construct a standard monomial basis for the space of sections H (Z; L), where Z is a BottSamelson variety and L a positive line bundle over Z. As a special case, we recover and complete the classical Standard Monomial Theory for an arbitrary semisimple algebraic group. ..."
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Cited by 16 (3 self)
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We construct a standard monomial basis for the space of sections H (Z; L), where Z is a BottSamelson variety and L a positive line bundle over Z. As a special case, we recover and complete the classical Standard Monomial Theory for an arbitrary semisimple algebraic group.
On fusion algebras and modular matrices
 Commun. Math. Phys
, 1999
"... Abstract: We consider the fusion algebras arising in e.g. WessZuminoWitten conformal field theories, affine KacMoody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix S, we find small sets of primary fields (equivalently, sets of high ..."
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Cited by 6 (2 self)
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Abstract: We consider the fusion algebras arising in e.g. WessZuminoWitten conformal field theories, affine KacMoody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix S, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the Ar fusion algebra at level k. We prove that for many choices of rank r and level k, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most r and k. We also find new, systematic sources of zeros in the modular matrix S. In addition, we obtain a formula relating the entries of S at fixed points, to entries of S at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of S, and by the fusion (Verlinde) eigenvalues.
The Space of Triangles, Vanishing Theorems, and Combinatorics
, 1996
"... We consider compactifications of (P \Delta ij , the space of triples of distinct points in projective space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schube ..."
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Cited by 5 (1 self)
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We consider compactifications of (P \Delta ij , the space of triples of distinct points in projective space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple.
Symplectic Multiple Flag Varieties Of Finite Type
 J. Algebra
"... this paper, we solve this problem for the symplectic group G = Sp 2n . We also give a complete enumeration of the orbits, and explicit representatives for them. The cases in our classification where one of the parabolics is a Borel subgroup, ..."
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Cited by 3 (0 self)
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this paper, we solve this problem for the symplectic group G = Sp 2n . We also give a complete enumeration of the orbits, and explicit representatives for them. The cases in our classification where one of the parabolics is a Borel subgroup,
SchurWeyl duality as an instrument of GaugeString duality,” arXiv:0804.2764 [hepth
"... Abstract. A class of mathematical dualities have played a central role in mapping states in gauge theory to states in the spacetime string theory dual. This includes the classical SchurWeyl duality between symmetric groups and Unitary groups, as well as generalisations involving Brauer and Hecke al ..."
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Abstract. A class of mathematical dualities have played a central role in mapping states in gauge theory to states in the spacetime string theory dual. This includes the classical SchurWeyl duality between symmetric groups and Unitary groups, as well as generalisations involving Brauer and Hecke algebras. The physical string dualities involved include examples from the AdS/CFT correspondence as well as the string dual of twodimensional Yang Mills.