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Title:

Stacky Lie Groups

Description:

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie grou...

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As an example, we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-03-27

Source:

http://www.maths.ed.ac.uk/~aar/blohmann.pdf

http://www.maths.ed.ac.uk/~aar/blohmann.pdf Minimize

Document Type:

text

Language:

en

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Preprint: IUB-TH-0412 Reconstruction of universal Drinfeld twists from representations

Description:

Universal Drinfeld twists are inner automorphisms which relate the coproduct of a quantum enveloping algebra to the coproduct of the undeformed enveloping algebra. Even though they govern the deformation theory of classical symmetries and have appeared in numerous applications, no twist for a semi-simple quantum enveloping algebra has ever been ...

Universal Drinfeld twists are inner automorphisms which relate the coproduct of a quantum enveloping algebra to the coproduct of the undeformed enveloping algebra. Even though they govern the deformation theory of classical symmetries and have appeared in numerous applications, no twist for a semi-simple quantum enveloping algebra has ever been computed. It is argued that universal twists can be reconstructed from their well known representations. A method to reconstruct an arbitrary element of the enveloping algebra from its irreducible representations is developed. For the twist this yields an algebra valued generating function to all orders in the deformation parameter, expressed by a combination of basic and ordinary hypergeometric functions. An explicit expression for the universal twist of su(2) is given up to third order. 1 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-06

Source:

http://arxiv.org/pdf/math/0410448v1.pdf

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text

Language:

en

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Title:

IUB-TP/2003-10 Perturbative Symmetries on Noncommutative Spaces

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Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semi-simple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity...

Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semi-simple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity of the action of the symmetry on the space. This implies that the noncommutative spaces considered can be realized as star products by particular ordering prescriptions which are compatible with the symmetry. These symmetry preserving ordering prescriptions are calculated for the quantum plane and four-dimensional quantum Euclidean space. Using these ordering prescriptions greatly facilitates the construction of invariant Lagrangians for quantum field theory on noncommutative spaces with a deformed symmetry. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-06

Source:

http://arxiv.org/pdf/math/0402200v1.pdf

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text

Language:

en

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Title:

Group-like objects in Poisson Geometry and algebra”, e-arXiv preprint, arXiv:math.SG/0701499

Description:

A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more general objects that can still be thought of as groups in many ways, such as quantum groups. We explain so...

A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more general objects that can still be thought of as groups in many ways, such as quantum groups. We explain some of the generalizations of groups which arise in Poisson geometry and quantization: the germ of a topological group, Poisson Lie groups, rigid monoidal structures on symplectic realizations, groupoids, 2-groups, stacky Lie groups, and hopfish algebras. 1 Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-13

Source:

http://arxiv.org/pdf/math/0701499v1.pdf

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text

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en

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Separation of noncommutative differential calculus on quantum Minkowski space

Separation of noncommutative differential calculus on quantum Minkowski space Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-14

Source:

http://arxiv.org/pdf/math/0506249v1.pdf

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text

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en

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Title:

STACKY LIE GROUPS

Description:

Abstract. Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stack...

Abstract. Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As example we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus. Dedicated to the memory of my friend Des Sheiham 1. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-13

Source:

http://arxiv.org/pdf/math/0702399v3.pdf

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text

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en

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Title:

Stacky Lie groups

Description:

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category (bicategory) of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. St...

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category (bicategory) of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as weak 2-group objects in this Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-14

Source:

http://arxiv.org/pdf/math/0702399v1.pdf

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text

Language:

en

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category

category Minimize

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Title:

STACKY LIE GROUPS

Description:

Abstract. Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category (bicategory) of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of pr...

Abstract. Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category (bicategory) of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as weak 2-group objects in this category. Introducing a graphic PROP notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. It is proved that the category of all actions of a stacky group G on a given stack X is equivalent to the category of group homomorphisms from G to the stacky automorphism group Aut(X). A stacky group action is weakly faithful if and only if the associated stacky group homomorphism is a weak monomorphism. This leads to Cayley’s theorem for stacky groups: Every stacky group G is naturally a subobject of Aut(G). 1. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-14

Source:

http://arxiv.org/pdf/math/0702399v2.pdf

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Document Type:

text

Language:

en

DDC:

512 Algebra *(computed)*

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Title:

Hopfish structure and modules over irrational rotation algebras.” Contemporary Mathematics (2006): preprint arXiv:math.QA/0604405

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Abstract. Inspired by the group structure on S 1 /Z, we introduce a weak hopfish structure on an irrational rotation algebra A of finite Fourier series. We consider a class of simple A-modules defined by invertible elements, and we compute the tensor product between these modules defined by the hopfish structure. This class of simple modules tur...

Abstract. Inspired by the group structure on S 1 /Z, we introduce a weak hopfish structure on an irrational rotation algebra A of finite Fourier series. We consider a class of simple A-modules defined by invertible elements, and we compute the tensor product between these modules defined by the hopfish structure. This class of simple modules turns out to generate an interesting commutative unital ring. 1. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-06

Source:

http://arxiv.org/pdf/math/0604405v2.pdf

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text

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en

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Title:

vorgelegt von

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The spin of particles on a non-commutative geometry is investigated within the framework of the representation theory of the q-deformed Poincaré algebra. An overview of the q-Lorentz algebra is given, including its representation theory with explicit formulas for the q-Clebsch-Gordan coefficients. The vectorial form of the q-Lorentz algebra (Wes...

The spin of particles on a non-commutative geometry is investigated within the framework of the representation theory of the q-deformed Poincaré algebra. An overview of the q-Lorentz algebra is given, including its representation theory with explicit formulas for the q-Clebsch-Gordan coefficients. The vectorial form of the q-Lorentz algebra (Wess), the quantum double form (Woronowicz), and the dual of the q-Lorentz group (Majid) are shown to be essentially isomorphic. The construction of q-Minkowski space and the q-Poincaré algebra is reviewed. The q-Euclidean sub-algebra, generated by rotations and translations, is studied in detail. The results allow for the construction of the q-Pauli-Lubanski vector, which, in turn, is used to determine the q-spin Casimir and the q-little algebras for both the massive and the massless case. Irreducible spin representations of the q-Poincaré algebra are constructed in an angular momentum basis, accessible to physical interpretation. It is shown how representations can be constructed, alternatively, by the method of induction. Reducible representations by q-Lorentz Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-11-06

Source:

http://arxiv.org/pdf/math/0110219v1.pdf

http://arxiv.org/pdf/math/0110219v1.pdf Minimize

Document Type:

text

Language:

en

DDC:

512 Algebra *(computed)*

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