o *ήc@s,ddlmZddlmZGdddeZdS))Standard_Cartan)eyec@sheZdZdZddZddZddZdd Zd d Zd d Z ddZ ddZ ddZ ddZ ddZdS)TypeAze This class contains the information about the A series of simple Lie algebras. ==== cCs|dkrtdt|d|S)Nzn cannot be less than 1A) ValueErrorr__new__)clsnr ?/tmp/pip-target-vg8gfxp4/lib/python/sympy/liealgebras/type_a.pyr sz TypeA.__new__cCs |jdS)zDimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("A4") >>> c.dimension() 5 rr selfr r r dimensions zTypeA.dimensioncCs(|j}dg|d}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_root s zTypeA.basic_rootcCs||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 A_n the ith simple root is the root which has a 1 in the ith position, a -1 in the (i+1)th position, and zeroes elsewhere. This method returns the ith simple root for the A series. Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("A4") >>> c.simple_root(1) [1, -1, 0, 0, 0] r)r)rrr r r simple_root.szTypeA.simple_rootcCsR|j}i}d}td|D]}t|d|dD]}|d7}|||||<qq |S)a This method generates all the positive roots of A_n. This is half of all of the roots of A_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 posrootskrrr r r positive_rootsKszTypeA.positive_rootscCs|d|jS)z9 Returns the highest weight root for A_n r)rr rr r r highest_rooteszTypeA.highest_rootcCs|j}||dS)z; Returns the total number of roots for A_n rr rr r r r rootsl z TypeA.rootscCsv|j}dt|}d}||dkr+d|||df<d|||df<|d7}||dksd|d<d||d|df<|S)a9 Returns the Cartan matrix for A_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('A4') >>> c.cartan_matrix() Matrix([ [ 2, -1, 0, 0], [-1, 2, -1, 0], [ 0, -1, 2, -1], [ 0, 0, -1, 2]]) rr)rr)r r)rr mrr r r cartan_matrixss   zTypeA.cartan_matrixcCs|j}|ddS)zE Returns the number of independent generators of A_n rrr rr r r basisrz TypeA.basiscCs|j}dt|ddS)z= Returns the Lie algebra associated with A_n zsu(r))r strrr r r lie_algebraszTypeA.lie_algebracCsN|j}dddtd|dDd}|dddtd|dD7}|S)Nz---css|]}dVqdS)0Nr .0rr r r sz'TypeA.dynkin_diagram..r z css|]}t|VqdS)N)r$r'r r r r)s)r joinr)rr diagr r r dynkin_diagrams""zTypeA.dynkin_diagramN)__name__ __module__ __qualname____doc__rrrrrrrr!r"r%r-r r r r rs! rN)sympy.liealgebras.cartan_typersympy.core.backendrrr r r r s