o *ήc• ã@s,ddlmZddlmZGdd„deƒZdS)é)ÚStandard_Cartané©ÚMatrixc@sLeZdZdd„Zdd„Zdd„Zdd„Zd d „Zd d „Zd d„Z dd„Z dS)ÚTypeGcCs|dkrtdƒ‚t |dd¡S)Néz n should be 2ÚG)Ú ValueErrorrÚ__new__)ÚclsÚn©r ú?/tmp/pip-target-vg8gfxp4/lib/python/sympy/liealgebras/type_g.pyr sz TypeG.__new__cCódS)zíDimension of the vector space V underlying the Lie algebra Examples ======== >>> from sympy.liealgebras.cartan_type import CartanType >>> c = CartanType("G2") >>> c.dimension() 3 ér ©Úselfr r rÚ dimensions zTypeG.dimensioncCs|dkrgd¢Sgd¢S)ašThe ith simple root of G_2 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("G2") >>> c.simple_root(1) [0, 1, -1] r©rréÿÿÿÿ©réþÿÿÿrr )rÚir r rÚ simple_rootszTypeG.simple_rootcCs.gd¢gd¢gd¢gd¢gd¢gd¢dœ}|S)aÛGenerate 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)rrr)rrr)rrr)rrr)rrrééér )rÚrootsr r rÚpositive_roots5s ÿzTypeG.positive_rootscCr)z; Returns the total number of roots of G_2" é r rr r rrJóz TypeG.rootscCstddgd¢ƒ}|S)aìThe Cartan matrix for G_2 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("G2") >>> c.cartan_matrix() Matrix([ [ 2, -1], [-3, 2]]) r)rréýÿÿÿrr)rÚmr r rÚ cartan_matrixPszTypeG.cartan_matrixcCr)zE Returns the number of independent generators of G_2 ér rr r rÚbasisgr z TypeG.basiscCsd}|S)Nu0≡<≡0 1 2r )rÚdiagr r rÚdynkin_diagrammszTypeG.dynkin_diagramN) Ú__name__Ú __module__Ú __qualname__r rrrrr#r%r'r r r rrs  rN)Ú cartan_typerÚsympy.core.backendrrr r r rÚs