Murray R. Bremner

Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications

First developed in the early 1980s by Lenstra, Lenstra, and Lovsz, the LLL algorithm was originally used to provide a polynomial-time algorithm for factoring polynomials with rational coefficients. It very quickly became an essential tool in integer linear programming problems and was later adapted for use in cryptanalysis. This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm. With numerous examples and suggested exercises, the text discusses various applications of lattice basis reduction to cryptography, number theory, polynomial factorization, and matrix canonical forms.

Special Identities for Quasi-Jordan Algebras

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities These identities are satisfied by the product ab = a ⊣ b + b ⊢ a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree ≤7 is a consequence of the three identities in degree ≤4, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.

Universal central extensions of elliptic affine Lie algebras

Let g be a simple complex (finite dimensional) Lie algebra, and let R be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form g⊗CR are considered; these generalize Kac–Moody loop algebras since for a curve of genus zero with two punctures R≂C[t,t−1]. The universal central extension of g⊗R is analogous to an untwisted affine Kac–Moody algebra. By Kassel’s theorem the kernel of the universal central extension is linearly isomorphic to the Kahler differentials of R modulo exact differentials. The dimension of the kernel for any R is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which ar...

Four-point affine Lie algebras

We consider Lie algebras of the form g X R where g is a simple complex Lie algebra and R = C[s, s-1, (s l)-, (s a)-l] for a E C {O, 1}. Atter showing that R is isomorphic to a quadratic extension of the ring C[t, t-1 l of Laurent polynomials, we prove that g 0 R is a quasi-graded Lie algebra with a triangular decomposition. We determine the universal central extension of g X R and show that the cocycles defining it are closely related to ultraspherical (Gegenbauer) polynomials.

LEIBNIZ TRIPLE SYSTEMS

We define Leibniz triple systems in a functorial manner using the algorithm of Kolesnikov and Pozhidaev which converts identities for algebras into identities for dialgebras. We verify that Leibniz triple systems are the natural analogues of Lie triple systems in the context of dialgebras by showing that both the iterated bracket in a Leibniz algebra and the permuted associator in a Jordan dialgebra satisfy the defining identities for Leibniz triple systems. We construct the universal Leibniz envelopes of Leibniz triple systems and prove that every identity satisfied by the iterated bracket in a Leibniz algebra is a consequence of the defining identities for Leibniz triple systems. To conclude, we present some examples of 2-dimensional Leibniz triple systems and their universal Leibniz envelopes.

How to compute the Wedderburn decomposition of a finite-dimensional associative algebra

Abstract This is a survey paper on algorithms that have been developed during the last 25 years for the explicit computation of the structure of an associative algebra of finite dimension over either a finite field or an algebraic number field. This constructive approach was initiated in 1985 by Friedl and Rónyai and has since been developed by Cohen, de Graaf, Eberly, Giesbrecht, Ivanyos, Küronya and Wales. I illustrate these algorithms with the case n = 2 of the rational semigroup algebra of the partial transformation semigroup PTn on n elements; this generalizes the full transformation semigroup and the symmetric inverse semigroup, and these generalize the symmetric group Sn .

On the Definition of Quasi-Jordan Algebra

Velásquez and Felipe recently introduced quasi-Jordan algebras based on the product in an associative dialgebra with operations ⊣ and ⊢. We determine the polynomial identities of degree ≤4 satisfied by this product. In addition to right commutativity and the right quasi-Jordan identity, we obtain a new associator-derivation identity.

Invariant Nonassociative Algebra Structures on Irreducible Representations of Simple Lie Algebras

An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra sl(2).

Identities for the Associator in Alternative Algebras

The associator is an alternating trilinear product for any alternative algebra. We study this trilinear product in three related algebras: the associator in a free alternative algebra, the associator in the Cayley algebra, and the ternary cross product on four-dimensional space. This last example is isomorphic to the ternary subalgebra of the Cayley algebra which is spanned by the non-quaternion basis elements. We determine the identities of degree ? 7 satisfied by these three ternary algebras. We discover two new identities in degree 7 satisfied by the associator in every alternative algebra and five new identities in degree 7 satisfied by the associator in the Cayley algebra. For the ternary cross product we recover the ternary derivation identity in degree 5 introduced by Filippov.

Jordan triple disystems

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm introduced by Bremner and Sanchez-Ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra. We apply the BSO algorithm to the Jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the KP algorithm; this provides a large class of examples of Jordan triple disystems. We formulate a general conjecture expressed by a commutative diagram relating the output of the KP and BSO algorithms. We conclude by generalizing the Jordan triple product in a Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to verify that resulting structures provide further examples of Jordan triple disystems. For this last result, we also provide an independent theoretical proof using Jordan structure theory.