On Algebras Which are Connected with the Semisimple Continuous Groups

Brauer, Richard

doi:10.2307/1968843

Cited by 545 publications

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“…The equality between this summation and the expression of Table 2 can be proved directly as in [12, p. 113], or by counting Brauer diagrams in two different ways, as is clear from [3]. For M = D n (n ≥ 4), the summation is…”

Section: Introductionmentioning

confidence: 94%

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Brauer algebras of simply laced type

2009

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Abstract. The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A n−1 on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph M , called the Brauer algebra of type M , and study its structure in the cases where M is a Coxeter graph of simply laced spherical type (so its connected components are of type A n−1 , Dn, E 6 , E 7 , E 8 ). We determine the representations and find the dimension. The algebra is semisimple and contains the group algebra of the Coxeter group of type M as a subalgebra. It is a ring homomorphic image of the Birman-MurakamiWenzl algebra of type M ; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.

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Section: Introductionmentioning

confidence: 94%

“…(ii) There is a unique linear representation, also denoted ρ B , of the Brauer algebra Br(M ) on V B extending the map ρ B of Lemma 3.4 with e i acting according to (3).…”

Section: Proof (I) Is Direct From the Definition Of The Action Of E mentioning

confidence: 99%

Brauer algebras of simply laced type

2009

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“…Brauer centralizer algebras were introduced in [8] in relation with the representation theory of the orthogonal and symplectic groups (see also [23]). Their structure was obtained by Wenzl in [22].…”

Section: Brauer Algebrasmentioning

confidence: 99%

“…recursive formula (8). This formula is further used to derive versions of the Wenzl formula corresponding to the quantum group specializations and to discuss the existence of idempotents in the non generic case.…”

Section: Introductionmentioning

confidence: 99%

Skein construction of idempotents in Birman-Murakami-Wenzl algebras

Beliakova¹,

Blanchet²

2001

Mathematische Annalen

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Abstract. We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas.

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“…The diagrammatical representation of the TL algebra is called the Brauer diagram [41] or Kauffman diagram [10] in literature. It is a planar (n, n) diagram including a "hidden" rectangle in the plane with 2n "hidden" distinct points: n points on its top edge and n points on its bottom edge, and they are connected by disjoint strings drawn in the rectangle.…”

Section: Diagrammatical Representation Of Tl Algebramentioning

confidence: 99%

Braid group, Temperley-Lieb algebra, and quantum information and computation

Zhang¹

2009

Contemporary Mathematics

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In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportation swapping and the virtual braid teleportation, respectively. Besides, we present a physical interpretation for the braid teleportation and explain it as a sort of crossed measurement. On the other hand, we propose the extended Temperley-Lieb diagrammatical approach to various topics including quantum teleportation, entanglement swapping, universal quantum computation, quantum information flow, and etc. The extended Temperley-Lieb diagrammatical rules are devised to present a diagrammatical representation for the extended Temperley-Lieb category which is the collection of all the Temperley-Lieb algebras with local unitary transformations. In this approach, various descriptions of quantum teleportation are unified in a diagrammatical sense, universal quantum computation is performed with the help of topological-like features, and quantum information flow is recast in a correct formulation. In other words, we propose the extended TemperleyLieb category as a mathematical framework to describe quantum information and computation involving maximally entangled states and local unitary transformations.

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On Algebras Which are Connected with the Semisimple Continuous Groups

Cited by 545 publications

References 0 publications

Brauer algebras of simply laced type

Brauer algebras of simply laced type

Skein construction of idempotents in Birman-Murakami-Wenzl algebras

Braid group, Temperley-Lieb algebra, and quantum information and computation

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