o *ήc@s,ddlmZddlmZGdddeZdS))Standard_Cartan)eyec@s\eZdZddZddZddZddZd d Zd d Zd dZ ddZ ddZ ddZ dS)TypeBcCs|dkrtdt|d|S)Nzn cannot be less than 2B) ValueErrorr__new__)clsnr ?/tmp/pip-target-vg8gfxp4/lib/python/sympy/liealgebras/type_b.pyr sz TypeB.__new__cCs|jS)zDimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("B3") >>> c.dimension() 3 r )selfr r r dimension s zTypeB.dimensioncCs dg|j}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)rijrootr r r basic_roots zTypeB.basic_rootcCs:|j}||kr||d|Sdg|j}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 B_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 n-th simple root is the root with a 1 in the nth position and zeroes elsewhere. This method returns the ith simple root for the B series. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("B3") >>> c.simple_root(2) [0, 1, -1] rr)r r)rrr rr r r simple_root%s   zTypeB.simple_rootcCs|j}i}d}td|dD]*}t|d|D] }|d7}|||||<|d7}|||}d||<|||<qqtd|D]}|d7}dg|}d||<|||<q>|S)a This method generates all the positive roots of A_n. This is half of all of the roots of B_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_rootsIs$    zTypeB.positive_rootscCs|j}d|dS)z< Returns the total number of roots for B_n" rrrr r r r rootsns z TypeB.rootscCs|j}dt|}d}||dkr+d|||df<d|||df<|d7}||dksd|d<d||d|df<d||d|df<|S)a9 Returns the Cartan matrix for B_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('B4') >>> c.cartan_matrix() Matrix([ [ 2, -1, 0, 0], [-1, 2, -1, 0], [ 0, -1, 2, -2], [ 0, 0, -1, 2]]) rrr)rr)r r)rr mrr r r cartan_matrixvs   zTypeB.cartan_matrixcCs|j}|d|dS)zE Returns the number of independent generators of B_n rrrr r r basissz TypeB.basiscCs|j}dtd|dS)z= Returns the Lie algebra associated with B_n zso(r))r strrr r r lie_algebraszTypeB.lie_algebracCsJ|j}dddtd|Dd}|dddtd|dD7}|S)Nz---css|]}dVqdS)0Nr .0rr r r sz'TypeB.dynkin_diagram..rz=>=0 z css|]}t|VqdS)N)r"r%r r r r's)r joinr)rr diagr r r dynkin_diagrams"zTypeB.dynkin_diagramN) __name__ __module__ __qualname__r rrrrrrr r#r*r r r r rs $%" rN) cartan_typersympy.core.backendrrr r r r s