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Alexander Schenkel - One of the best experts on this subject based on the ideXlab platform.
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Algebraic field theory operads and linear quantization
Letters in Mathematical Physics, 2019Co-Authors: Simen Bruinsma, Alexander SchenkelAbstract:We generalize the operadic approach to algebraic quantum field theory ( arXiv:1709.08657 ) to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose Left Adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories.
Tomasz Brzezinski - One of the best experts on this subject based on the ideXlab platform.
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the structure of corings induction functors maschke type theorem and frobenius and galois type properties
Algebras and Representation Theory, 2002Co-Authors: Tomasz BrzezinskiAbstract:Given a ring A and an A-coring C, we study when the forgetful functor from the category of right C-comodules to the category of right A-modules and its right Adjoint −⊗ A C are separable. We then proceed to study when the induction functor −⊗ A C is also the Left Adjoint of the forgetful functor. This question is closely related to the problem when A→ A Hom(C,A) is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of A fixed under the coaction of C is an equivalence. We also comment on possible dualisation of the notion of a coring.
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the structure of corings induction functors maschke type theorem and frobenius and galois type properties
arXiv: Rings and Algebras, 2000Co-Authors: Tomasz BrzezinskiAbstract:Given a ring $A$ and an $A$-coring $\cC$ we study when the forgetful functor from the category of right $\cC$-comodules to the category of right $A$-modules and its right Adjoint $-\otimes_A\cC$ are separable. We then proceed to study when the induction functor $-\otimes_A\cC$ is also the Left Adjoint of the forgetful functor. This question is closely related to the problem when $A\to {}_A{\rm Hom}(\cC,A)$ is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of $A$ fixed under the coaction of $\cC$ is an equivalence. We also comment on possible dualisation of the notion of a coring.
Simen Bruinsma - One of the best experts on this subject based on the ideXlab platform.
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Algebraic field theory operads and linear quantization
Letters in Mathematical Physics, 2019Co-Authors: Simen Bruinsma, Alexander SchenkelAbstract:We generalize the operadic approach to algebraic quantum field theory ( arXiv:1709.08657 ) to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose Left Adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories.
Mahanta Snigdhayan - One of the best experts on this subject based on the ideXlab platform.
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C∗–algebraic drawings of dendroidal sets
'Mathematical Sciences Publishers', 2019Co-Authors: Mahanta SnigdhayanAbstract:In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. We introduce the concept of a C*-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on C*-algebras. We show that the construction is functorial and, in fact, it is the Left Adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable infinity-categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant K-theory. Finally, a method to analyze graph algebras in terms of trees is sketched
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$C^*$-algebraic drawings of dendroidal sets
'Mathematical Sciences Publishers', 2019Co-Authors: Mahanta SnigdhayanAbstract:In recent years the theory of dendroidal sets has emerged as an important framework for higher algebra. In this article we introduce the concept of a $C^*$-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an object in the category of presheaves on $C^*$-algebras. We show that the construction is functorial and, in fact, it is the Left Adjoint of a Quillen adjunction between combinatorial model categories. We use this construction to produce a bridge between the two prominent paradigms of noncommutative geometry via adjunctions of presentable $\infty$-categories, which is the primary motivation behind this article. As a consequence we obtain a single mechanism to construct bivariant homology theories in both paradigms. We propose a (conjectural) roadmap to harmonize algebraic and analytic (or topological) bivariant K-theory. Finally, a method to analyse graph algebras in terms of trees is sketched.Comment: 28 pages; v2 expanded version with some improvements; v3 revised and added references; v4 some changes according to the suggestions of the referees (to appear in Algebr. Geom. Topol.
Vladimir Dotsenko - One of the best experts on this subject based on the ideXlab platform.
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a quillen adjunction between algebras and operads koszul duality and the lagrange inversion formula
arXiv: Category Theory, 2016Co-Authors: Vladimir DotsenkoAbstract:We define, for a somewhat standard forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operad" functors, the right inverse and the Left Adjoint of the forgetful functor. Those functors turn out to be related by operadic Koszul duality, and that relationship can be utilised to provide examples showing limitations of two standard tools of the Koszul duality theory. We also apply these functors to get a homotopical algebra proof of the Lagrange inversion formula.
