o *ήcV@s,ddlmZddlmZGdddeZdS))Standard_Cartan)eyec@s\eZdZddZddZddZddZd d Zd d Zd dZ ddZ ddZ ddZ dS)TypeDcCs|dkrtdt|d|S)Nzn cannot be less than 3D) ValueErrorr__new__)clsnr ?/tmp/pip-target-vg8gfxp4/lib/python/sympy/liealgebras/type_d.pyr sz TypeD.__new__cCs|jS)zDmension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("D4") >>> c.dimension() 4 r )selfr r r dimension s zTypeD.dimensioncCs$|j}dg|}d||<d||<|S)z This is a method just to generate roots with a 1 iin the ith position and a -1 in the jth position. rrr)rijr rootr r r basic_roots  zTypeD.basic_rootcCsD|j}||kr||d|Sdg|}d||d<d||d<|S)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. In D_n, the first n-1 simple roots are the same as the roots in A_(n-1) (a 1 in the ith position, a -1 in the (i+1)th position, and zeroes elsewhere). The nth simple root is the root in which there 1s in the nth and (n-1)th positions, and zeroes elsewhere. This method returns the ith simple root for the D series. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("D4") >>> c.simple_root(2) [0, 1, -1, 0] rr)r r)rrr rr r r simple_root(s   zTypeD.simple_rootcCsv|j}i}d}td|dD]*}t|d|D] }|d7}|||||<|d7}|||}d||<|||<qq|S)a This method generates all the positive roots of A_n. This is half of all of the roots of D_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]} rr)r ranger)rr posrootskrrrr r r positive_rootsOs  zTypeD.positive_rootscCs|j}d||dS)z< Returns the total number of roots for D_n" rrrrr r r r rootsmsz TypeD.rootscCs|j}dt|}d}||dkr+d|||df<d|||df<|d7}||dksd||d|df<d||d|df<d||d|df<d|d<|S)aM Returns the Cartan matrix for D_n. 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('D4') >>> c.cartan_matrix() Matrix([ [ 2, -1, 0, 0], [-1, 2, -1, -1], [ 0, -1, 2, 0], [ 0, -1, 0, 2]]) rrrr)rr)r r)rr mrr r r cartan_matrixus   zTypeD.cartan_matrixcCs|j}||ddS)zE Returns the number of independent generators of D_n rrrrr r r basissz TypeD.basiscCs|j}dtd|dS)z> Returns the Lie algebra associated with D_n" zso(r))r strrr r r lie_algebraszTypeD.lie_algebracCs|j}d|dt|dd}|d|dd7}|d|dd7}|d|dd7}|ddd td|Dd7}|d d d td|dDd t|7}|S) Nz rr z0 z| z---css|]}dVqdS)0Nr .0rr r r sz'TypeD.dynkin_diagram..z css|]}t|VqdS)N)r"r&r r r r(s)r r"joinr)rr diagr r r dynkin_diagrams".zTypeD.dynkin_diagramN) __name__ __module__ __qualname__r rrrrrrr r#r+r r r r rs'# rN) cartan_typersympy.core.backendrrr r r r s