o *ήc4&@s0ddlmZddlmZmZGdddeZdS))Standard_Cartan)eyeRationalc@sTeZdZddZddZddZddZd d Zd d Zd dZ ddZ ddZ dS)TypeEcCs&|dks|dkr tdt|d|S)NzInvalid value of nE) ValueErrorr__new__)clsnr?/tmp/pip-target-vg8gfxp4/lib/python/sympy/liealgebras/type_e.pyr sz TypeE.__new__cCsdS)zDimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("E6") >>> c.dimension() 8 rr)selfrrr dimension s zTypeE.dimensioncCsdgd}d||<d||<|S)z This is a method just to generate roots with a -1 in the ith position and a 1 in the jth position. rrrr)rijrootrrr basic_roots zTypeE.basic_rootcCs|j}|dkrdgd}d|d<d|d<|S|dkr)dgd}d|d<d|d<|S|dvr5|d kr5td |dkrA|dkrAtd ||d |dS) a Every lie algebra has a unique root system. Given a root system Q, there is a subset of the roots such that an element of Q is called a simple root if it cannot be written as the sum of two elements in Q. If we let D denote the set of simple roots, then it is clear that every element of Q can be written as a linear combination of elements of D with all coefficients non-negative. This method returns the ith simple root for E_n. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("E6") >>> c.simple_root(2) [1, 1, 0, 0, 0, 0, 0, 0] rgrg?r)rrrzE6 only has six simple roots!zE7 has only 7 simple roots!)r r r)rrr rrrr simple_root's   zTypeE.simple_rootc Cs|j}|dkri}d}t|dD].}t|d|dD]"}|d7}|||}|||<|d7}|||}d||<|||<qqtddtddtddtddtddtddtddtddg}tddD]o}tddD]g}tddD]_} tddD]W} tddD]O} ||| | | ddkr|d7}|dkrtdd|d<|dkrtdd|d<| dkrtdd|d<| dkrtdd|d<| dkrtdd|d<|||<qq|quqnqg|S|dkri}d}t|dD].}t|d|dD]"}|d7}|||}|||<|d7}|||}d||<|||<qq|d7}gd ||<tddtddtddtddtddtddtddtddg}tddD]}tddD]}tddD]} tddD]v} tddD]m} tddD]d} ||| | | | ddkr|d7}|dkrtdd|d<|dkrtdd|d<| dkrtdd|d<| dkrtdd|d<| dkrtdd|d<| dkrtdd|d <|||<qkqdq]qVqOqH|S|d kri}d}t|D].}t|d|D]#}|d7}|||}|||<|d7}|||}d||<|||<qqtddtddtddtddtddtddtddtddg}tddD]}tddD]}tddD]} tddD]} tddD]} tddD]{} tddD]r} ||| | | | | ddkr|d7}|dkrtdd|d<|dkrtdd|d<| dkrtdd|d<| dkrtdd|d<| dkrtdd|d<| dkrtdd|d <| dkrtdd|d<|||<qiqbq[qTqMqFq?|Sd S) a This method generates all the positive roots of A_n. This is half of all of the roots of E_n; by multiplying all the positive roots by -1 we get the negative roots. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("A3") >>> c.positive_roots() {1: [1, -1, 0, 0], 2: [1, 0, -1, 0], 3: [1, 0, 0, -1], 4: [0, 1, -1, 0], 5: [0, 1, 0, -1], 6: [0, 0, 1, -1]} rrrrrrr)rrrrrrrrrN)r rangerr)rr posrootskrrrabcdefgrrrpositive_rootsOs   (      ("           ( &       zTypeE.positive_rootscC.|j}|dkr dS|dkrdS|dkrdSdS)z: Returns the total number of roots of E_n rHr~rNr rr rrrrootsz TypeE.rootscCs|j}dt|}d}||dkr+d|||df<d|||df<|d7}||dksd|d<|d<d|d<|d<d|d <d||d|df<|S) a9 Returns the Cartan matrix for G_2 The Cartan matrix matrix for a Lie algebra is generated by assigning an ordering to the simple roots, (alpha[1], ...., alpha[l]). Then the ijth entry of the Cartan matrix is (). Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType('A4') >>> c.cartan_matrix() Matrix([ [ 2, -1, 0, 0], [-1, 2, -1, 0], [ 0, -1, 2, -1], [ 0, 0, -1, 2]]) rrrr)rr)rr)rr)rr)rr)r r)rr mrrrr cartan_matrixs   zTypeE.cartan_matrixcCr()zE Returns the number of independent generators of E_n rNrrNr,r-rrrbasis r/z TypeE.basisc Csz|j}dtdd}|d7}|d7}|d7}|dddtd |Dd7}|d d d dtd |d D7}|S)Nz r z 0 z | z---css|]}dVqdS)0Nr.0rrrr sz'TypeE.dynkin_diagram..rz1 z css|]}t|VqdS)N)strr8rrrr:sr)r r;joinr)rr diagrrrdynkin_diagrams"&zTypeE.dynkin_diagramN) __name__ __module__ __qualname__r rrrr'r.r1r5r>rrrrrs ( % rN) cartan_typersympy.core.backendrrrrrrrs