o *ήcA@s0ddlmZddlmZmZGdddeZdS))Standard_Cartan)MatrixRationalc@sTeZdZddZddZddZddZd d Zd d Zd dZ ddZ ddZ dS)TypeFcCs|dkrtdt|ddS)Nz n should be 4F) ValueErrorr__new__)clsnr ?/tmp/pip-target-vg8gfxp4/lib/python/sympy/liealgebras/type_f.pyr sz TypeF.__new__cCdS)zDimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("F4") >>> c.dimension() 4 rr selfr r r dimension s zTypeF.dimensioncCs$|j}dg|}d||<d||<|S)zFGenerate roots with 1 in ith position and -1 in jth position rr)r )rijr rootr r r basic_roots  zTypeF.basic_rootcCsV|dkr ||d|S|dkrdgd}d|d<|S|dkr)tddgd}|SdS)aThe ith simple root of F_4 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. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("F4") >>> c.simple_root(3) [0, 0, 0, 1] rrrrN)rr)rrrr r r simple_root&s zTypeF.simple_rootcCs|j}i}d}td|dD]*}t|d|D] }|d7}|||||<|d7}|||}d||<|||<qqtd|D]}|d7}dg|}d||<|||<q>|d7}tddg|}|||<tddD]}|d7}tddg|}tdd||<|||<qgtddtddtddtddg||d<tddtddtddtddg||d<tddtddtddtddg||d<tddtddtddtddg||d<|S)aGenerate all the positive roots of A_n This is half of all of the roots of F_4; 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]} rrrrrr)r rangerr)rr posrootskrrrr r rpositive_rootsFs<     ,,,,zTypeF.positive_rootscCr)z; Returns the total number of roots for F_4 0r rr r rrootszz TypeF.rootscCstddgd}|S)a'The Cartan matrix for F_4 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]]) r)rrrrrrrrrrrrrrr)r)rmr r r cartan_matrixszTypeF.cartan_matrixcCr)zE Returns the number of independent generators of F_4 4r rr r rbasisr!z TypeF.basiscCs&d}|dddtddD7}|S)Nz0---0=>=0---0 z css|]}t|VqdS)N)str).0rr r r sz'TypeF.dynkin_diagram..r)joinr)rdiagr r rdynkin_diagramszTypeF.dynkin_diagramN) __name__ __module__ __qualname__r rrrrr r$r&r-r r r rrs  4 rN) cartan_typersympy.core.backendrrrr r r rs