o *ήcW@s,ddlmZddlmZGdddeZdS))Standard_Cartan)eyec@s\eZdZddZddZddZddZd d Zd d Zd dZ ddZ ddZ ddZ dS)TypeCcCs|dkrtdt|d|S)Nzn cannot be less than 3C) ValueErrorr__new__)clsnr ?/tmp/pip-target-vg8gfxp4/lib/python/sympy/liealgebras/type_c.pyr sz TypeC.__new__cCs |j}|S)zDimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("C3") >>> c.dimension() 3 r selfr r r r dimension s zTypeC.dimensioncCs$|j}dg|}d||<d||<|S)zGGenerate roots with 1 in ith position and a -1 in jth position rrr)rijr rootr r r basic_roots  zTypeC.basic_rootcCs:|j}||kr||d|Sdg|j}d||d<|S)aThe ith simple root for the C series Every lie algebra has a unique root system. Given a root system Q, there is a subset of the roots such that an element of Q is called a simple root if it cannot be written as the sum of two elements in Q. If we let D denote the set of simple roots, then it is clear that every element of Q can be written as a linear combination of elements of D with all coefficients non-negative. In C_n, the first n-1 simple roots are the same as the roots in A_(n-1) (a 1 in the ith position, a -1 in the (i+1)th position, and zeroes elsewhere). The nth simple root is the root in which there is a 2 in the nth position and zeroes elsewhere. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("C3") >>> c.simple_root(2) [0, 1, -1] rr)r r)rrr rr r r simple_root#s   zTypeC.simple_rootcCs|j}i}d}td|dD]*}t|d|D] }|d7}|||||<|d7}|||}d||<|||<qqtd|D]}|d7}dg|}d||<|||<q>|S)aGenerates all the positive roots of A_n This is half of all of the roots of C_n; by multiplying all the positive roots by -1 we get the negative roots. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("A3") >>> c.positive_roots() {1: [1, -1, 0, 0], 2: [1, 0, -1, 0], 3: [1, 0, 0, -1], 4: [0, 1, -1, 0], 5: [0, 1, 0, -1], 6: [0, 0, 1, -1]} rrr)r ranger)rr posrootskrrrr r r positive_rootsHs$    zTypeC.positive_rootscCs|j}d|dS)z< Returns the total number of roots for C_n" rrrr r r rootsms z TypeC.rootscCsv|j}dt|}d}||dkr+d|||df<d|||df<|d7}||dksd|d<d||d|df<|S)a(The Cartan matrix for C_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('C4') >>> c.cartan_matrix() Matrix([ [ 2, -1, 0, 0], [-1, 2, -1, 0], [ 0, -1, 2, -1], [ 0, 0, -2, 2]]) rrr)rr)r r)rr mrr r r cartan_matrixus   zTypeC.cartan_matrixcCs|j}|d|dS)zE Returns the number of independent generators of C_n rrrrr r r basissz TypeC.basiscCs|j}dtd|dS)z> Returns the Lie algebra associated with C_n" zsp(r))r strrr r r lie_algebraszTypeC.lie_algebracCsJ|j}dddtd|Dd}|dddtd|dD7}|S)Nz---css|]}dVqdS)0Nr .0rr r r sz'TypeC.dynkin_diagram..rz=<=0 z css|]}t|VqdS)N)r#r&r r r r(s)r joinr)rr diagr r r dynkin_diagrams"zTypeC.dynkin_diagramN) __name__ __module__ __qualname__r rrrrrr r!r$r+r r r r rs %%" rN) cartan_typersympy.core.backendrrr r r r s