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41-584: [REDACTED] Look up bn  or .bn in Wiktionary, the free dictionary. BN , Bn or bn may refer to: Businesses and organizations [ edit ] Arts and media [ edit ] RTV BN , a Bosnian Serb broadcaster Bandai Namco , a gaming and entertainment conglomerate Barnes & Noble , an American chain of bookstores BN (band) , Belarusian rock band Transport [ edit ] Braniff International Airways (IATA code BN),

82-436: A Russian nuclear reactor class Batch normalization , in artificial intelligence Benzyl functional group (Bn), in organic chemistry Billion (disambiguation) (bn), a number Boron nitride , a chemical compound Bulimia nervosa , an eating disorder Dynkin diagram B n , in mathematical analysis Other uses [ edit ] BN (biscuit) , a Franco-British brand of baked food Bachelor of Nursing ,

123-533: A base. Since these two roots are at angle of 120 degrees (with a length ratio of 1), the Dynkin diagram consists of two vertices connected by a single edge: [REDACTED] [REDACTED] [REDACTED] . Dynkin diagrams must satisfy certain constraints; these are essentially those satisfied by finite Coxeter–Dynkin diagrams , together with an additional crystallographic constraint. Dynkin diagrams are closely related to Coxeter diagrams of finite Coxeter groups , and

164-455: A former American airline in service from 1928 to 1982 Britten-Norman , a British manufacturer, based on the Isle of Wight, producing Islander and Trislander aircraft La Brugeoise et Nivelles , a Belgian railway rolling stock manufacturer, now part of Alstom Burlington Northern Railroad , a United States railroad that operated from 1970 to 1996 Horizon Airlines (Australia) (IATA code BN),

205-607: A former Australian airline, ending service in 2004 Other businesses and organizations [ edit ] Banca Nuova , an Italian bank Banque Nationale (disambiguation) several banks Barisan Nasional (also known as "National Front"), a political coalition in Malaysia British Naturism , the UK naturist society Groupe Danone (uronext stock exchange code BN), a French food-products multinational Military [ edit ] Bangladesh Navy Battalion ,

246-432: A graph with one vertex for each element of Δ {\displaystyle \Delta } . Then insert edges between each pair of vertices according to the following recipe. If the roots corresponding to the two vertices are orthogonal, there is no edge between the vertices. If the angle between the two roots is 120 degrees, we put one edge between the vertices. If the angle is 135 degrees, we put two edges, and if

287-682: A large unit Places [ edit ] Countries [ edit ] Brunei (ISO 3166: BN), Southeast Asia Bahrain (WMO: BN), West Asia Benin (FIPS 10-4: BN), West Africa Bosnia and Herzegovina (LOC MARC: BN), Europe Regions [ edit ] BN postcode area , Sussex, England Bandarban District , Bangladesh Province of Benevento , Italy Bloomington-Normal , Illinois, US Place of worship [ edit ] Baitun Nur Mosque , Calgary, Alberta, Canada Science, technology and mathematics [ edit ] .bn , Brunei's top-level Internet domain BN-reactor ,

328-413: A length ratio of 3 {\displaystyle {\sqrt {3}}} . (There are no edges when the roots are orthogonal, regardless of the length ratio.) In the A 2 {\displaystyle A_{2}} root system, shown at right, the roots labeled α {\displaystyle \alpha } and β {\displaystyle \beta } form

369-464: A professional degree Bacon Number , an actor's professional proximity to Kevin Bacon Bengali language (ISO 639: bn), spoken in eastern South Asia Billion (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title BN . If an internal link led you here, you may wish to change the link to point directly to

410-400: A simple interpretation in terms of mathematical objects of interest. There are natural maps down – from Dynkin diagrams to undirected Dynkin diagrams; respectively, from root systems to the associated Weyl groups – and right – from undirected Dynkin diagrams to Coxeter diagrams; respectively from Weyl groups to finite Coxeter groups. The down map is onto (by definition) but not one-to-one, as

451-508: A simply-laced diagram. The one condition on the automorphism for folding to be possible is that distinct nodes of the graph in the same orbit (under the automorphism) must not be connected by an edge; at the level of root systems, roots in the same orbit must be orthogonal. At the level of diagrams, this is necessary as otherwise the quotient diagram will have a loop, due to identifying two nodes but having an edge between them, and loops are not allowed in Dynkin diagrams. The nodes and edges of

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492-501: A symmetry (satisfying one condition, below) can be quotiented by the symmetry, yielding a new, generally multiply laced diagram, with the process called folding (due to most symmetries being 2-fold). At the level of Lie algebras, this corresponds to taking the invariant subalgebra under the outer automorphism group, and the process can be defined purely with reference to root systems, without using diagrams. Further, every multiply laced diagram (finite or infinite) can be obtained by folding

533-1021: Is a Weyl group. Dynkin diagrams are conventionally numbered so that the list is non-redundant: n ≥ 1 {\displaystyle n\geq 1} for A n , {\displaystyle A_{n},} n ≥ 2 {\displaystyle n\geq 2} for B n , {\displaystyle B_{n},} n ≥ 3 {\displaystyle n\geq 3} for C n , {\displaystyle C_{n},} n ≥ 4 {\displaystyle n\geq 4} for D n , {\displaystyle D_{n},} and E n {\displaystyle E_{n}} starting at n = 6. {\displaystyle n=6.} The families can however be defined for lower n, yielding exceptional isomorphisms of diagrams, and corresponding exceptional isomorphisms of Lie algebras and associated Lie groups. Trivially, one can start

574-474: Is a single non-trivial automorphism (Out = C 2 , the cyclic group of order 2), while for D 4 , the automorphism group is the symmetric group on three letters ( S 3 , order 6) – this phenomenon is known as " triality ". It happens that all these diagram automorphisms can be realized as Euclidean symmetries of how the diagrams are conventionally drawn in the plane, but this is just an artifact of how they are drawn, and not intrinsic structure. For A n ,

615-405: Is an order 2 automorphism of B 2 ≅ C 2 {\displaystyle \mathrm {B} _{2}\cong \mathrm {C} _{2}} and of F 4 , while in characteristic 3 there is an order 2 automorphism of G 2 . But doesn't apply in all circumstances: for example, such automorphisms need not arise as automorphisms of the corresponding algebraic group, but rather on

656-487: Is disconnected, and the automorphism corresponds to switching the two nodes. For D 4 , the fundamental representation is isomorphic to the two spin representations, and the resulting symmetric group on three letter ( S 3 , or alternatively the dihedral group of order 6, Dih 3 ) corresponds both to automorphisms of the Lie algebra and automorphisms of the diagram. The automorphism group of E 6 corresponds to reversing

697-862: Is how the dual representation acts. For D n , the diagram automorphism is switching the two nodes at the end of the Y, and corresponds to switching the two chiral spin representations . Realized as the Lie algebra s o 2 n , {\displaystyle {\mathfrak {so}}_{2n},} the outer automorphism can be expressed as conjugation by a matrix in O(2 n ) with determinant −1. When n = 3, one has D 3 ≅ A 3 , {\displaystyle \mathrm {D} _{3}\cong \mathrm {A} _{3},} so their automorphisms agree, while D 2 ≅ A 1 × A 1 {\displaystyle \mathrm {D} _{2}\cong \mathrm {A} _{1}\times \mathrm {A} _{1}}

738-490: Is standardized, Coxeter diagram and group notation is varied and sometimes agrees with Dynkin diagram notation and sometimes does not. Lastly, sometimes associated objects are referred to by the same notation, though this cannot always be done regularly. Examples include: These latter notations are mostly used for objects associated with exceptional diagrams – objects associated to the regular diagrams (A, B, C, D) instead have traditional names. The index (the n ) equals to

779-421: Is that they classify semisimple Lie algebras over algebraically closed fields . One classifies such Lie algebras via their root system , which can be represented by a Dynkin diagram. One then classifies Dynkin diagrams according to the constraints they must satisfy, as described below. Dropping the direction on the graph edges corresponds to replacing a root system by the finite reflection group it generates,

820-506: Is the finite reflection group associated to the root system. Thus B n may refer to the unoriented diagram (a special kind of Coxeter diagram), the Weyl group (a concrete reflection group), or the abstract Coxeter group. Although the Weyl group is abstractly isomorphic to the Coxeter group, a specific isomorphism depends on an ordered choice of simple roots. Likewise, while Dynkin diagram notation

861-404: Is used to refer to all such interpretations, depending on context; this ambiguity can be confusing. The central classification is that a simple Lie algebra has a root system, to which is associated an (oriented) Dynkin diagram; all three of these may be referred to as B n , for instance. The un oriented Dynkin diagram is a form of Coxeter diagram , and corresponds to the Weyl group, which

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902-568: The B n and C n diagrams map to the same undirected diagram, with the resulting Coxeter diagram and Weyl group thus sometimes denoted BC n . The right map is simply an inclusion – undirected Dynkin diagrams are special cases of Coxeter diagrams, and Weyl groups are special cases of finite Coxeter groups – and is not onto, as not every Coxeter diagram is an undirected Dynkin diagram (the missed diagrams being H 3 , H 4 and I 2 ( p ) for p  = 5  p  ≥ 7), and correspondingly not every finite Coxeter group

943-585: The E n family . In addition to isomorphism between different diagrams, some diagrams also have self-isomorphisms or " automorphisms ". Diagram automorphisms correspond to outer automorphisms of the Lie algebra, meaning that the outer automorphism group Out = Aut/Inn equals the group of diagram automorphisms. The diagrams that have non-trivial automorphisms are A n ( n > 1 {\displaystyle n>1} ), D n ( n > 1 {\displaystyle n>1} ), and E 6 . In all these cases except for D 4 , there

984-421: The crystallographic restriction theorem , and that Coxeter diagrams are undirected, while Dynkin diagrams are (partly) directed. The corresponding mathematical objects classified by the diagrams are: The blank in the upper right, corresponding to directed graphs with underlying undirected graph any Coxeter diagram (of a finite group), can be defined formally, but is little-discussed, and does not appear to admit

1025-436: The undirected graph. For precision, in this article "Dynkin diagram" will mean directed, and the underlying undirected graph will be called an "undirected Dynkin diagram". Then Dynkin diagrams and Coxeter diagrams may be related as follows: By this is meant that Coxeter diagrams of finite groups correspond to point groups generated by reflections, while Dynkin diagrams must satisfy an additional restriction corresponding to

1066-552: The Lie algebra. For example, B 4 {\displaystyle B_{4}} corresponds to s o 2 ⋅ 4 + 1 = s o 9 , {\displaystyle {\mathfrak {so}}_{2\cdot 4+1}={\mathfrak {so}}_{9},} which naturally acts on 9-dimensional space, but has rank 4 as a Lie algebra. The simply laced Dynkin diagrams, those with no multiple edges (A, D, E) classify many further mathematical objects; see discussion at ADE classification . For example,

1107-406: The angle is 150 degrees, we put three edges. (These four cases exhaust all possible angles between pairs of positive simple roots. ) Finally, if there are any edges between a given pair of vertices, we decorate them with an arrow pointing from the vertex corresponding to the longer root to the vertex corresponding to the shorter one. (The arrow is omitted if the roots have the same length.) Thinking of

1148-425: The arrow as a "greater than" sign makes it clear which way the arrow should go. Dynkin diagrams lead to a classification of root systems. The angles and length ratios between roots are related . Thus, the edges for non-orthogonal roots may alternatively be described as one edge for a length ratio of 1, two edges for a length ratio of 2 {\displaystyle {\sqrt {2}}} , and three edges for

1189-405: The diagram automorphism is reversing the diagram, which is a line. The nodes of the diagram index the fundamental weights , which (for A n −1 ) are ⋀ i C n {\displaystyle \bigwedge ^{i}C^{n}} for i = 1 , … , n {\displaystyle i=1,\dots ,n} , and the diagram automorphism corresponds to

1230-478: The diagram, and can be expressed using Jordan algebras . Disconnected diagrams, which correspond to semi simple Lie algebras, may have automorphisms from exchanging components of the diagram. In positive characteristic there are additional "diagram automorphisms" – roughly speaking, in characteristic p one is sometimes allowed to ignore the arrow on bonds of multiplicity p in the Dynkin diagram when taking diagram automorphisms. Thus in characteristic 2 there

1271-550: The duality ⋀ i C n ↦ ⋀ n − i C n . {\displaystyle \bigwedge ^{i}C^{n}\mapsto \bigwedge ^{n-i}C^{n}.} Realized as the Lie algebra s l n + 1 , {\displaystyle {\mathfrak {sl}}_{n+1},} the outer automorphism can be expressed as negative transpose, T ↦ − T T {\displaystyle T\mapsto -T^{\mathrm {T} }} , which

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1312-547: The edge in G 2 points from the class of the 3 outer nodes (valence 1), to the class of the central node (valence 3). The foldings of finite diagrams are: Similar foldings exist for affine diagrams, including: The notion of foldings can also be applied more generally to Coxeter diagrams – notably, one can generalize allowable quotients of Dynkin diagrams to H n and I 2 ( p ). Geometrically this corresponds to projections of uniform polytopes . Notably, any simply laced Dynkin diagram can be folded to I 2 ( h ), where h

1353-477: The families at n = 0 {\displaystyle n=0} or n = 1 , {\displaystyle n=1,} which are all then isomorphic as there is a unique empty diagram and a unique 1-node diagram. The other isomorphisms of connected Dynkin diagrams are: These isomorphisms correspond to isomorphism of simple and semisimple Lie algebras, which also correspond to certain isomorphisms of Lie group forms of these. They also add context to

1394-949: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=BN&oldid=1258187754 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages bn">bn The requested page title contains unsupported characters : ">". Return to Main Page . Dynkin diagram The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed , in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected , in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named. The fundamental interest in Dynkin diagrams

1435-439: The level of points valued in a finite field. Diagram automorphisms in turn yield additional Lie groups and groups of Lie type , which are of central importance in the classification of finite simple groups. The Chevalley group construction of Lie groups in terms of their Dynkin diagram does not yield some of the classical groups, namely the unitary groups and the non- split orthogonal groups . The Steinberg groups construct

1476-409: The number of nodes in the diagram, the number of simple roots in a basis, the dimension of the root lattice and span of the root system, the number of generators of the Coxeter group, and the rank of the Lie algebra. However, n does not equal the dimension of the defining module (a fundamental representation ) of the Lie algebra – the index on the Dynkin diagram should not be confused with the index on

1517-457: The quotient ("folded") diagram are the orbits of nodes and edges of the original diagram; the edges are single unless two incident edges map to the same edge (notably at nodes of valence greater than 2) – a "branch point" of the map, in which case the weight is the number of incident edges, and the arrow points towards the node at which they are incident – "the branch point maps to the non-homogeneous point". For example, in D 4 folding to G 2 ,

1558-435: The so-called Weyl group , and thus undirected Dynkin diagrams classify Weyl groups. They have the following correspondence for the Lie algebras associated to classical groups over the complex numbers: For the exceptional groups, the names for the Lie algebra and the associated Dynkin diagram coincide. Dynkin diagrams can be interpreted as classifying many distinct, related objects, and the notation "A n , B n , ..."

1599-470: The symbol A 2 {\displaystyle A_{2}} may refer to: Consider a root system , assumed to be reduced and integral (or "crystallographic"). In many applications, this root system will arise from a semisimple Lie algebra . Let Δ {\displaystyle \Delta } be a set of positive simple roots . We then construct a diagram from Δ {\displaystyle \Delta } as follows. Form

1640-421: The terminology is often conflated. Dynkin diagrams differ from Coxeter diagrams of finite groups in two important respects: A further difference, which is only stylistic, is that Dynkin diagrams are conventionally drawn with double or triple edges between nodes (for p  = 4, 6), rather than an edge labeled with " p ". The term "Dynkin diagram" at times refers to the directed graph, at times to

1681-565: The unitary groups A n , while the other orthogonal groups are constructed as D n , where in both cases this refers to combining a diagram automorphism with a field automorphism. This also yields additional exotic Lie groups E 6 and D 4 , the latter only defined over fields with an order 3 automorphism. The additional diagram automorphisms in positive characteristic yield the Suzuki–Ree groups , B 2 , F 4 , and G 2 . A (simply-laced) Dynkin diagram (finite or affine ) that has

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