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Constructing Homomorphism Spaces and Endomorphism Rings

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Constructing Homomorphism Spaces and Endomorphism Rings View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Symbolic Computation (2001) 32, 101{117 doi:10.1006/jsco.2001.0454 Available online at http://www.idealibrary.com on Constructing Homomorphism Spaces and Endomorphism Rings EDWARD L. GREENy, LENWOOD S. HEATHz AND CRAIG A. STRUBLEz yDepartment of Mathematics, Virginia Tech, Blacksburg, VA 24061, U.S.A. zDepartment of Computer Science, Virginia Tech, Blacksburg, VA 24061, U.S.A. We present a new deterministic algorithm for constructing homomorphism spaces and endomorphism rings of finite-dimensional modules. The modules are given via vertex projective presentations over path algebras and finite-dimensional quotients of path algebras. We use the theory of right Gr¨obnerbases to encode modules and to construct appropriate systems of equations for computing homomorphism spaces and endomor- phism rings. The algorithm is implemented in the computer algebra system GAP and is included in Hopf, a computational package for noncommutative algebra. The per- formance of our implementation for computing endomorphism rings is experimentally compared with the implementation in Magma for the same class of modules. These experiments show that our implementation has a better time complexity. c 2001 Academic Press 1. Introduction Computing the structure of modules is a fundamental task in the representation theory of finite-dimensional algebras. Computer algebra systems such as GAP (The GAP Group, Achen, St Andrews, 1999) and Magma ( Bosma et al., 1994) include procedures for determining the structure of modules. Another important tool in the study of modules is the MeatAxe of Parker (1984) with improvements by Holt and Rees (1994). The primary function of the MeatAxe is to find proper submodules of modules over finite algebras. These systems have been used extensively in the study of finite groups. We present an algorithm for computing the endomorphism ring of a module that is useful in decomposing modules. Let K be a field, let Λ be a finite-dimensional algebra, and let M be a finite-dimensional right Λ-module. The endomorphism ring EndΛ(M) of M is the ring of all endomorphisms f : M M. It is well known that decompositions of M into nontrivial submodules can be determined! from decompositions of the identity map IdM in EndΛ(M) as direct sums of pairwise orthogonal idempotents. Chistov et al. (1997, Section 4) demonstrate how to compute a complete set of pairwise orthogonal idempotents in EndΛ(M). Consequently, the efficient construction of endomorphism rings is important to studying the structure of modules. Schneider (1990) appears to have produced the first implementations of algorithms for constructing endomorphism rings. These implementations are included in the computer algebra system Cayley, the predecessor to Magma. His initial implementation for con- structing endomorphism rings uses the direct approach. Observe that a module M is also a K-vector space of dimension d. The actions of the generators of Λ can be written as 0747{7171/01/070101 + 17 $35.00/0 c 2001 Academic Press 102 E. L. Green et al. d d matrices with entries in K. If g is a generator of Λ, let fg be the matrix describing the× action of g on M. The ring of d d matrices X that satisfy the system of equations × X fg fg X = 0; · − · for all generators g of Λ, is isomorphic to EndΛ(M). In the same paper, Schneider improves on this approach for group algebras by reducing the number of unknowns and the number of equations to solve by randomly choosing endomorphisms and refining approximations of the solution space. The process iterates, using information gained from the random endomorphism, until the entire solution space is determined. Leedham-Green implemented an unpublished algorithm for computing endomorphism rings, which is included in Magma ( Bosma et al., 1994). Smith includes an implementation of a randomized algorithm in the AutAG share package included in GAP version 3.4.4 (Sch¨onert et al., 1995). The algorithm is tuned for modules over group algebras. The comments in the source code state that the algorithm is based on discussions with Leedham-Green, Lux, and Niemeyer, but that its performance is not as good as the implementation in Magma. Sz}oke (1998) presents a randomized algorithm using peakwords (Lux et al., 1994) to construct the regular representation of EndΛ(M). That algorithm is included in the C implementation of the MeatAxe package (Ringe, 2000). According to Sz}oke's dissertation, her algorithm generally performs better than the algorithm in Magma. In contrast, our algorithm is a deterministic algorithm instead of a randomized al- gorithm. Furthermore, an input module is described by a projective presentation over path algebras instead of describing the generator actions as K-matrices. Path algebras naturally arise in the study of finite-dimensional modules, which is explained further in Section 3. A module described by a projective presentation is readily encoded as linear actions of generators of a path algebra as well, which is discussed in Section 5. Thus, previ- ous approaches can be used on modules given by a projective presentation, as long as they do not assume that the actions are invertible. Furthermore, there exists an algorithm to construct a projective presentation of a finite-dimensional module described by linear ac- tions for a finite number of generators. This algorithm is included in a forthcoming paper. We have implemented our endomorphism ring algorithm in Hopf ( Green et al., 1999), which will be released as a share package for GAP. The performance of our algorithm is comparable to the performance of the algorithm in Magma. We elaborate on the performance in Section 7. Earlier versions of this work appear in Green et al. (2000) and Struble (2000). The work extends that earlier work by generalizing the construction to homomorphism spaces. This generalization is remarkably straightforward and should demonstrate the same performance benefits already seen in the construction of endomorphism rings. The remainder of this paper is structured as follows. In Section 2, we recall the general result of adjoint associativity and explicitly describe its application to homomorphism spaces and endomorphism rings. We review path algebras and Gr¨obnerbases in Section 3. Relevant results for right modules defined over path algebras and right Gr¨obnerbases are contained in Section 4. Linear actions of path algebras on right modules are described in Section 5. Section 6 combines the results in the previous sections, explicitly describing the construction of homomorphism spaces and endomorphism rings. The performance of the GAP implementation of our algorithm is discussed in Section 7. Section 8 contains concluding remarks and open problems. Constructing Homomorphism Spaces 103 2. Adjoint Associativity We begin by recalling a general result found in Hungerford (1974, p. 214). Theorem 2.1. [Hungerford, Theorem 5.10, Adjoint Associativity] Let R and S be rings, and let AR, RBS, CS be (bi)-modules. Then there is an isomorphism of Abelian groups Υ : HomS(A R B; C) = HomR(A; HomS(B; C)); ⊗ ∼ defined for each f : A R B C by ⊗ ! [(Υf)(a)](b) = f(a b): ⊗ If A, B, and C are as given in Theorem 2.1, then A R B is a right S-module via ⊗ (a b)s = a (bs). We also have that HomS(B; C) is a right R-module given by the ⊗ ⊗ left R-module structure of B:(gr)(b) = g(rb) for r R; b B; and g HomS(B; C). Checking that the map (gr) is a right R-module homomorphism2 2 is left as2 an exercise. Our particular interest in endomorphism rings is a specific application of this general theorem, as we now demonstrate. Let K be a field, let Λ be a K-algebra, and let M and N be finite-dimensional right Λ-modules. Since M and N are K-vector spaces, M and N are both left and right K-modules. The dual of N, denoted N ∗, is HomK(N; K), the set of K-linear maps from N to K. The dual N ∗ is a left Λ-module via (λh)(n) = h(nλ); (1) where λ Λ; h HomK(N; K); and n N. Hence, N ∗ is a left Λ-module and a right 2 2 2 K-module with λ(hα) = (λh)α, where λ Λ, h N ∗, and α K. 2 2 2 Since N is finite dimensional, there exists a natural isomorphism between HomK(N ∗; K) and N. In particular, let n1; : : : ; nd be a K-basis of N. Define maps n∗ : N K in f g i ! N ∗ by 0 if i = j; n∗(nj) = ( 6 i 1 if i = j: The set n1∗; : : : ; nd∗ forms the dual basis of N ∗. Define Ψ : HomK(N ∗; K) N by Ψ(h) = d f g ! h(n∗)ni. One can check that the inverse of Ψ is the map ρ : N HomK(N ∗; K) Pi=1 i ! given by ρ(n): N ∗ K where ρ(n)(h) = h(n) for h N ∗. Thus, Ψ is an isomorphism. Note that ρ is independent! of the choice of basis for2 N, so it follows that Ψ is also independent of the choice of basis, even though it is defined with a choice of basis. Theorem 2.2. Let K be a field, let Λ be a K-algebra, and let M and N be finite- dimensional right Λ-modules. Then there exists an isomorphism of Abelian groups Φ : HomK(M Λ N ∗; K) = HomΛ(M; N); ⊗ ∼ d where Φ is given by [(Φf)(m)] = f(m n∗)ni. Pi=1 ⊗ i Proof. Applying Theorem 2.1, we obtain the existence of the isomorphism Υ : HomK(M Λ N ∗; K) = HomΛ(M; HomK(N ∗; K)); ⊗ ∼ 104 E.
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