The 2 and 3 series are called twisted affine diagrams. They are usually labeled by their order of symmetry, with order-3 implied with no label. Retrieved 29 June 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.
Population below 6 years was 28, Scheduled Castes numbered 38, As per the census, Nandigram I block had a total population of ,, out of which 88, were males and 85, were females. Nandigram I block registered a population growth of Decadal growth for the combined Midnapore district was As per census, literacy in Purba Medinipur district was See also — List of West Bengal districts ranked by literacy rate.
Bengali is the local language in these areas. In census Hindus numbered , and formed Muslims numbered 70, and formed Others numbered and formed 0. Nandigram I CD Block registered There were 57 fertiliser depots, 12 seed stores and 31 fair price shops in the CD Block.
The agricultural sector is the lifeline of a predominantly rural economy. In many cases the canals are drainage canals which get the backflow of river water at times of high tide or the rainy season. The average size of land holding in Purba Medinipur, in , was 0. In , the total area irrigated in Nandigram I CD Block was 2, hectares, out of which 2, hectares were irrigated by tank water and hectares by other means.
In net area sown in Nandigram I CD Block was 13, hectares and the area in which more than one crop was grown was 4, hectares. Although the Bargadari Act of recognised the rights of bargadars to a higher share of crops from the land that they tilled, it was not implemented fully.
Large tracts, beyond the prescribed limit of land ceiling, remained with the rich landlords. From onwards major land reforms took place in West Bengal. Land in excess of land ceiling was acquired and distributed amongst the peasants. In , Nandigram I CD Block produced 44, tonnes of Aman paddy , the main winter crop, from 23, hectares, 3, tonnes of Boro paddy, the spring crop, from hectares, 6, tonnes of Aus paddy, the summer crop, from 3, hectares, 41 tonnes of jute from 3 hectares and 7, tonnes of potatoes from hectares.
It also produced pulses and oil seeds. Betelvine is a major source of livelihood in Purba Medinipur district, particularly in Tamluk and Contai subdivisions. Betelvine production in was the highest amongst all the districts and was around a third of the total state production. In , Purba Mednipur produced 2, tonnes of cashew nuts from 3, hectares of land.
The nearest railway station is 23 km from the block headquarters. The ferry services are available every 20 minutes, . In , Nandigram I CD Block had primary schools with 9, students, 10 middle schools with students, 10 high schools with 8, students and 14 higher secondary schools with 16, students. Nandigram I CD Block had 1 general college with 1, students and institutions for special and non-formal education with 18, students. Sitananda College at Nandigram was established in It offers courses in arts and science.
In , Nandigram I CD Block had 1 block primary health centre and 2 primary health centres , with total 46 beds and 9 doctors excluding private bodies. It had 30 family welfare sub centres. From Wikipedia, the free encyclopedia. Community development block in West Bengal, India. To Nandigram via Singur". Retrieved 8 July Frontlines of Revolutionary Struggle, March Retrieved 10 July Retrieved 29 June Retrieved 6 July Archived from the original on 29 July Retrieved 6 November Directorate of Census Operations, West Bengal, Retrieved 9 November Purba Medinipur - Revised in March In addition to isomorphism between different diagrams, some diagrams also have self-isomorphisms or " automorphisms ".
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 , the diagram automorphism is reversing the diagram, which is a line. 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. 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 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.
But doesn't apply in all circumstances: 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 the unitary groups 2 A n , while the other orthogonal groups are constructed as 2 D n , where in both cases this refers to combining a diagram automorphism with a field automorphism.
This also yields additional exotic Lie groups 2 E 6 and 3 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 , 2 B 2 , 2 F 4 , and 2 G 2. A simply-laced Dynkin diagram finite or affine that has 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.
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. The nodes and edges of 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 , 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: 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 is the Coxeter number , which corresponds geometrically to projection to the Coxeter plane.
Folding can be applied to reduce questions about semisimple Lie algebras to questions about simply-laced ones, together with an automorphism, which may be simpler than treating multiply laced algebras directly; this can be done in constructing the semisimple Lie algebras, for instance. Folding by Automorphisms for further discussion. Some additional maps of diagrams have meaningful interpretations, as detailed below.
However, not all maps of root systems arise as maps of diagrams. For example, there are two inclusions of root systems of A 2 in G 2 , either as the six long roots or the six short roots. However, the nodes in the G 2 diagram correspond to one long root and one short root, while the nodes in the A 2 diagram correspond to roots of equal length, and thus this map of root systems cannot be expressed as a map of the diagrams.
Some inclusions of root systems can be expressed as one diagram being an induced subgraph of another, meaning "a subset of the nodes, with all edges between them". This is because eliminating a node from a Dynkin diagram corresponds to removing a simple root from a root system, which yields a root system of rank one lower. By contrast, removing an edge or changing the multiplicity of an edge while leaving the nodes unchanged corresponds to changing the angles between roots, which cannot be done without changing the entire root system.
Thus, one can meaningfully remove nodes, but not edges. Removing a node from a connected diagram may yield a connected diagram simple Lie algebra , if the node is a leaf, or a disconnected diagram semisimple but not simple Lie algebra , with either two or three components the latter for D n and E n. At the level of Lie algebras, these inclusions correspond to sub-Lie algebras. The maximal subgraphs are as follows; subgraphs related by a diagram automorphism are labeled "conjugate":.
Finally, duality of diagrams corresponds to reversing the direction of arrows, if any: A Dynkin diagram with no multiple edges is called simply laced , as are the corresponding Lie algebra and Lie group. In this case the Dynkin diagrams exactly coincide with Coxeter diagrams, as there are no multiple edges.
Dynkin diagrams classify complex semisimple Lie algebras. Real semisimple Lie algebras can be classified as real forms of complex semisimple Lie algebras, and these are classified by Satake diagrams , which are obtained from the Dynkin diagram by labeling some vertices black filled , and connecting some other vertices in pairs by arrows, according to certain rules.
Dynkin diagrams are named for Eugene Dynkin , who used them in two papers , simplifying the classification of semisimple Lie algebras;  see Dynkin When Dynkin left the Soviet Union in , which was at the time considered tantamount to treason, Soviet mathematicians were directed to refer to "diagrams of simple roots" rather than use his name.
Undirected graphs had been used earlier by Coxeter to classify reflection groups , where the nodes corresponded to simple reflections; the graphs were then used with length information by Witt in reference to root systems, with the nodes corresponding to simple roots, as they are used today.
Beyond simplicity, a further benefit of this convention is that diagram automorphisms are realized by Euclidean isometries of the diagrams. There are also conventions about numbering the nodes. The most common modern convention had developed by the s and is illustrated in Bourbaki Dynkin diagrams are equivalent to generalized Cartan matrices , as shown in this table of rank 2 Dynkin diagrams with their corresponding 2 x 2 Cartan matrices.
A multi-edged diagram corresponds to the nondiagonal Cartan matrix elements -a 21 , -a 12 , with the number of edges drawn equal to max -a 21 , -a 12 , and an arrow pointing towards nonunity elements. The Cartan matrix determines whether the group is of finite type if it is a Positive-definite matrix , i.
The indefinite type often is further subdivided, for example a Coxeter group is Lorentzian if it has one negative eigenvalue and all other eigenvalues are positive. Moreover, multiple sources refer to hyberbolic Coxeter groups, but there are several non-equivalent definitions for this term.
In the discussion below, hyperbolic Coxeter groups are a special case of Lorentzian, satisfying an extra condition. Note that for rank 2, all negative determinant Cartan matrices correspond to hyperbolic Coxeter group. But in general, most negative determinant matrices are neither hyperbolic nor Lorentzian. These are usually not applied to finite and affine graphs.
For undirected groups, Coxeter diagrams are interchangeable. They are usually labeled by their order of symmetry, with order-3 implied with no label. Many multi-edged groups can be obtained from a higher ranked simply-laced group by applying a suitable folding operation. There are extensions of Dynkin diagrams, namely the affine Dynkin diagrams ; these classify Cartan matrices of affine Lie algebras.
These are classified in Kac , Chapter 4, pp. The 2 and 3 series are called twisted affine diagrams. See Dynkin diagram generator for diagrams. Here are all of the Dynkin graphs for affine groups up to 10 nodes. Other directed-graph variations are given with a superscript value 2 or 3 , representing foldings of higher order groups.
These are categorized as Twisted affine diagrams. The set of compact and noncompact hyperbolic Dynkin graphs has been enumerated. Compact hyperbolic Dynkin diagrams exist up to rank 5, and noncompact hyperbolic graphs exist up to rank Edit Read in another language Dynkin diagram.
Finite Dynkin diagrams Affine extended Dynkin diagrams Contents. The root lattice generated by the root system, as in the E 8 lattice.