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Van Daele Alfons - One of the best experts on this subject based on the ideXlab platform.
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A class of C*-algebraic locally compact quantum groupoids Part II: Main theory
'Elsevier BV', 2019Co-Authors: Kahng Byung-jay, Van Daele AlfonsAbstract:This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical Idempotent Element $E$ plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map.Comment: Some minor revisions made; Bibliography update
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A class of C*-algebraic locally compact quantum groupoids Part I. Motivation and definition
2017Co-Authors: Kahng Byung-jay, Van Daele AlfonsAbstract:In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the C*-algebra framework. Existence of a certain canonical Idempotent Element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass
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Weak Multiplier Hopf Algebras. The main theory
2012Co-Authors: Van Daele Alfons, Wang ShuanhongAbstract:A weak multiplier Hopf algebra is a pair (A,\Delta) of a non-degenerate Idempotent algebra A and a coproduct $\Delta$ on A. The coproduct is a coassociative homomorphism from A to the multiplier algebra M(A\otimes A) with some natural extra properties (like the existence of a counit). Further we impose extra but natural conditions on the ranges and the kernels of the canonical maps T_1 and T_2 defined from A\otimes A to M(A\otimes A) by T_1(a\otimes b)=\Delta(a)(1\otimes b) and T_2(a\ot b)=(a\otimes 1)\Delta(b). The first condition is about the ranges of these maps. It is assumed that there exists an Idempotent Element E\in M(A\otimes A) such that \Delta(A)(1\ot A)=E(A\ot A) and (A\otimes 1)\Delta(A)=(A\otimes A)E. The second condition determines the behavior of the coproduct on the legs of E. We require (\Delta\otimes \iota)(E)=(\iota\otimes\Delta)(E)=(1\otimes E)(E\ot 1)=(E\otimes 1)(1\otimes E) where $\iota$ is the identity map and where $\Delta\otimes \iota$ and $\iota\otimes\Delta$ are extensions to the multipier algebra M(A\otimes A). Finally, the last condition determines the kernels of the canonical maps T_1 and T_2 in terms of this Idempotent E by a very specific relation. From these conditions we develop the theory. In particular, we construct a unique antipode satisfying the expected properties and various other data. Special attention is given to the regular case (that is when the antipode is bijective) and the case of a *-algebra (where regularity is automatic). Weak Hopf algebras are special cases of such weak multiplier Hopf algebras. Conversely, if the underlying algebra of a (regular) weak multiplier Hopf algebra has an identity, it is a weak Hopf algebra. Also any groupoid, finite or not, yields two weak multiplier Hopf algebras in duality
R B Zhang - One of the best experts on this subject based on the ideXlab platform.
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the second fundamental theorem of invariant theory for the orthogonal group
Annals of Mathematics, 2012Co-Authors: Gustav I. Lehrer, R B ZhangAbstract:Let V = C n be endowed with an orthogonal form and G = O(V ) be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism : Br(n)! EndG(V r ), where Br(n) is the r-string Brauer algebra with parameter n. However the kernel of has remained elusive. In this paper we show that, in analogy with the case of GL(V ), for r n + 1, has a kernel which is generated by a single Idempotent Element E, and we give a simple explicit formula for E. Using the theory of cellular algebras, we show how E may be used to determine the multiplicities of the irreducible representations of O(V ) in V r . We also show how our results extend to the case where C is replaced by an appropriate eld of positive characteristic, and we comment on quantum analogues of our results.
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the second fundamental theorem of invariant theory for the orthogonal group
arXiv: Group Theory, 2011Co-Authors: Gustav I. Lehrer, R B ZhangAbstract:Let $V=\C^n$ be endowed with an orthogonal form and $G=\Or(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $\nu:B_r(n)\to\End_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$ has remained elusive. In this paper we show that, in analogy with the case of $\GL(V)$, for $r\geq n+1$, $\nu$ has kernel which is generated by a single Idempotent Element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\Or(V)$ in $V^{\ot r}$. We also show how our results extend to the case where $\C$ is replaced by an appropriate field of positive characteristic, and comment on quantum analogues of our results.
Benjamin Bohme - One of the best experts on this subject based on the ideXlab platform.
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multiplicativity of the Idempotent splittings of the burnside ring and the g sphere spectrum
Advances in Mathematics, 2019Co-Authors: Benjamin BohmeAbstract:Abstract We provide a complete characterization of the equivariant commutative ring structures of all the factors in the Idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any Idempotent Element, which is of independent interest to group theorists. As an application, we obtain an explicit description of the incomplete sets of norm functors which are present in the Idempotent splitting of the equivariant stable homotopy category.
Gustav I. Lehrer - One of the best experts on this subject based on the ideXlab platform.
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the second fundamental theorem of invariant theory for the orthogonal group
Annals of Mathematics, 2012Co-Authors: Gustav I. Lehrer, R B ZhangAbstract:Let V = C n be endowed with an orthogonal form and G = O(V ) be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism : Br(n)! EndG(V r ), where Br(n) is the r-string Brauer algebra with parameter n. However the kernel of has remained elusive. In this paper we show that, in analogy with the case of GL(V ), for r n + 1, has a kernel which is generated by a single Idempotent Element E, and we give a simple explicit formula for E. Using the theory of cellular algebras, we show how E may be used to determine the multiplicities of the irreducible representations of O(V ) in V r . We also show how our results extend to the case where C is replaced by an appropriate eld of positive characteristic, and we comment on quantum analogues of our results.
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the second fundamental theorem of invariant theory for the orthogonal group
arXiv: Group Theory, 2011Co-Authors: Gustav I. Lehrer, R B ZhangAbstract:Let $V=\C^n$ be endowed with an orthogonal form and $G=\Or(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $\nu:B_r(n)\to\End_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$ has remained elusive. In this paper we show that, in analogy with the case of $\GL(V)$, for $r\geq n+1$, $\nu$ has kernel which is generated by a single Idempotent Element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\Or(V)$ in $V^{\ot r}$. We also show how our results extend to the case where $\C$ is replaced by an appropriate field of positive characteristic, and comment on quantum analogues of our results.
Kahng Byung-jay - One of the best experts on this subject based on the ideXlab platform.
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A class of C*-algebraic locally compact quantum groupoids Part II: Main theory
'Elsevier BV', 2019Co-Authors: Kahng Byung-jay, Van Daele AlfonsAbstract:This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids ("quantum groupoids of separable type"), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical Idempotent Element $E$ plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map.Comment: Some minor revisions made; Bibliography update
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A class of C*-algebraic locally compact quantum groupoids Part I. Motivation and definition
2017Co-Authors: Kahng Byung-jay, Van Daele AlfonsAbstract:In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the C*-algebra framework. Existence of a certain canonical Idempotent Element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass
