| Exterior algebras, which are defined in a vector space V , are a class of veryimportant algebra. Exterior algebras and their modules, with strong applicationbackground. There is a series of study on exterior algebras and their modules re-cently, however, the extension of two modules is an elementary and very interestingpart of the study of modules.Let k is an algebraically closed field, V be an 3-dimensional linear space overk, V be the exterior algebra over V , a,b,c be linearly independent vectors inV . In this paper, we make e?orts to researching on the linear extension andisomorphism of two linear modules Mand Lof their representation matrices areFm(a,b) and Tnt(a,b,c).We still apply the method of representation matrix in this paper. Firstly, wecalculate the representation matrix of linear extension modules, and thus discussterms of isomorphism.Let N is the linear extension module of M with the help of L, its represen-1.If m > n and M,L,D1,D2 be defined as lemma 3.1. C1 = C11 + C12andC2 = C21 +C22 in which all the elements in C11 and C21 are belong to L(a,b) andall the elements in C12 and C22 are belong to L(c). We can properly choose thebases of Pi(M) Pi(L)(i = 0,1,2) such that1)All the elements in C11 and C21 are belong to L(a) and there exist matrices If N1and N2 are linear extension modules of M with the help of L, their repre-sentation matrices arehave decompositions as theorem 3.2, let C12 = (zi1j2 c)n×m and C 12 =(zi j1 2c)n×m, we have:2.Let M,L,C12,C 12 defined as above. N1 and N2 are linear extension modulesof M with the help of L. When m > n, if there exist nonzero elements e1 and ei inthe field we have an isomorphism between N1 and N2. |
