Timothy J. Hodges

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• PhD
• Professor Emeritus at University of Cincinnati

In this article we discuss the Belavin-Drinfeld quantum groups, one of the most fascinating and elusive classes of non-standard quantum groups. Belavin and Drinfeld classified all non-skewsymmetric classical r r -matrices for simple Lie algebras g \mathfrak {g} . Etingof, Schedler and Schiffmann constructed explicit formal quantizations of the asso...

Semi-regular sequences over F2 are sequences of homogeneous elements of the algebra B(n)=F2[X1,...,Xn]/(X12,...,Xn2), which have a given Hilbert series and can be thought of as having as few relations between them as possible. It is believed that most such systems are semi-regular and this property has important consequences for understanding the c...

Semi-regular sequences over F2 are sequences of homogeneous elements of the algebra B(n)=F2[X1,...,Xn]/(X12,...,Xn2), which have as few relations between them as possible. It is believed that most such systems are F2-semi-regular and this property has important consequences for understanding the complexity of Gröbner basis algorithms such as F4 and...

Semi-regular sequences over are sequences of homogeneous elements of the algebra , which have as few relations between them as possible. They were introduced in order to assess the complexity of Gröbner basis algorithms such as and for the solution of polynomial equations. Despite the experimental evidence that semi-regular sequences are common, it...

Semi-regular sequences over F 2 are sequences of homogeneous elements of the algebra B (n) = F_2 [X_1 , . . . , X_n]/(X^2_1 , . . . , X^2_n), which have as few relations between them as possible. They were introduced in order to assess the complexity of Gröbner basis algorithms such as F4, F5 for the solution of polynomial equations. Despite the ex...

Polynomial systems arising from a Weil descent have many applications in cryptography, including the HFE cryptosystem and the elliptic curve discrete logarithm problem over small characteristic fields. Understanding the exact complexity of solving these systems is essential for the applications. A first step in that direction is to study the first...

Let K be the field GF(3). We calculate the growth of the ideal Aλ where A is the algebra of functions from Kⁿ → Kⁿ and λ is a quadratic function. Specifically we calculate dim Akλ where Ak is the space of polynomials of degree less than or equal to k. This question arises in the analysis of the complexity of Gröbner basis attacks on multivariate qu...

Let F be a finite field of odd characteristic q. We calculate the degree of regularity for a quadratic element of the algebra F[X1,…,Xn]/(X1q,…,Xnq), the first degree at which non-trivial annihilation occurs. We also give an explicit formula for the dimension of the space of non-trivial annihilators of a given degree. These questions arise in multi...

In this paper, we present and prove the first closed formula bounding the degree of regularity of an HFE system over an arbitrary finite field. Though these bounds are not necessarily optimal, they can be used to deduce 1) if D, the degree of the corresponding HFE polynomial, and q, the size of the corresponding finite field, are fixed, inverting H...

High-density lipoproteins (HDLs) mediate cholesterol transport and protection from cardiovascular disease. Although synthetic HDLs have been studied for 30 years, the structures of human plasma-derived HDL and its major protein apolipoprotein apoA-I are unknown. We separated normal human HDL into five density subfractions and then further isolated...

We give exact formulas for the growth of the ideal Aλ for λ a quadratic element of the algebra of Boolean functions over the Galois field GF(2). That is, we calculate \(\dim A_k \lambda\) where A k is the subspace of elements of degree less than or equal to k. These results clarify some of the assertions made in the article of Yang, Chen and Courto...

We show that each triangular Poisson Lie group can be decomposed into Poisson submanifolds each of which is a quotient of a symplectic manifold. The Marsden-Weinstein-Meyer symplectic reduction technique is then used to give a complete description of the symplectic foliation of all triangular Poisson structures on Lie groups. The results are illust...

The Matsumoto-Imai (MI) cryptosystem was the first mul- tivariate public key cryptosystem proposed for practical use. Though MI is now considered insecure due to Patarin's linearization attack, the core idea of MI has been used to construct many variants such as Sflash, which has recently been accepted for use in the New European Schemes for Signat...

A Tamed Transformation Method (TTM) cryptosystem was proposed by T. T. Moh in 1999. We describe how the first implementation scheme of the TTM system can be defeated. The computational complexity of our attack is 2³³ computations on the finite field with 2⁸ elements. The cipher of the TTM systems are degree 2 polynomial maps derived from compositio...

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa-Ueno. We define a two-parameter family of generalizations of the Shibukawa-Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted a...

Since its genesis in the early 1980s, the subject of quantum groups has grown rapidly. By the late 1990s most of the foundational issues had been resolved and many of the outstanding problems clearly formulated. To take stock and to discuss the most fruitful directions for future research many of the world's leading figures in this area met at the...

Let G be a connected, simply connected Poisson-Lie group with quasitriangular Lie bialgebra g. An explicit description of the double D(g) is given, together with the embeddings of g and g^*. This description is then used to provide a construction of the double D(G). The aim of this work is to describe D(G) in sufficient detail to be able to apply t...

The coefficients of certain operators on V x V can be constructed using generating functions. Necessary and sufficient conditions are given for some such operators to satisfy the Yang-Baxter equation. As a corollary we obtain a simple, direct proof that the Cremmer-Gervais R-matrices satisfy the Yang-Baxter equation. This approach also clarifies Cr...

An explicit quantization is given of certain skew-symmetric solutions of the classical Yang–Baxter equation, yielding a family of R-matrices which generalize to higher dimensions the Jordanian R-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa–Ueno, Yang–Baxter operators on meromorphi...

An explicit quantization is given of certain skew-symmetric solutions of the classical Yang-Baxter, yielding a family of $R$-matrices which generalize to higher dimensions the Jordanian $R$-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa-Ueno Yang-Baxter operators on meromorphic func...

In [4] Joseph proved the classication of the primitive ideals of the quantum group C q [G ] conjectured in [2]. We prove this result taking account of Joseph's analysis in [4] and of the methods already developed in [3].

The coefficients of certain operators on $V\otimes V$ can be constructed using generating functions. Necessary and sufficient conditions are given for some such operators to satisfy the Yang-Baxter equation. As a corollary we obtain a simple, direct proof that the Cremmer-Gervais R-matrices satisfy the Yang-Baxter equation. This approach also clari...

A direct proof is given of the fact that the Cremmer-Gervais R-matrices satisfy the Yang-Baxter equation.

A construction is given of a family of non-standard quantizations of the algebra of functions on a connected complex semi-simple algebraic group. For each ``disjoint'' triple in the sense of Belavin and Drinfeld, a 2-cocycle is constructed on certain multi-parameter quantum groups. The new non-standard quantum groups are the Hopf algebras obtained...

Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).

Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).

The double quantum groups q[D(G)] = q[G] ⋈ q[G] are the Hopf algebras underlying the complex quantum groups of which the simplest example is the quantum Lorentz group. They are nonstandard quantizations of the double groupG × G. We construct a corresponding quantized universal enveloping algebraUq(()) and prove that the pairing between q[D(G)] andU...

Multi-parameter versions U_p(g) and C_p[G] of the standard quantum groups U_q(g) and C_q[G] are considered where G is a semi-simple connected complex algebraic group and g is the Lie algebra of G. The primitive spectrum of C_p[G] is calculated, generalizing a result of Joseph for the standard quantum groups. This classification is compared with the...

this paper.1. Poisson Lie Groups1.1. Notation. Let g be a complex semi-simple Lie algebra associated to a Cartan matrix [a ij ] 16i;j6n .Let fd i g 16i6n be relatively prime positive integers such that [d i a ij ] 16i;j6n is symmetric positive denite.Let h be a Cartan subalgebra of g, R the associated root system, B = f 1 ; : : : ; n g a basis of R...

this paper. 1. Poisson Lie Groups 1.1. Notation. Let g be a complex semi-simple Lie algebra associated to a Cartan matrix [a ij ] 16i;j6n . Let fd i g 16i6n be relatively prime positive integers such that [d i a ij ] 16i;j6n is symmetric positive denite. Let h be a Cartan subalgebra of g, R the associated root system, B = f 1 ; : : : ; n g a basis...

The double quantum groups are the Hopf algebras underlying the complex quantum groups of which the simplest example is the quantum Lorentz group. They are non- standard quantizations of the double group $G \times G$. We construct a corresponding quantized universal enveloping algebra (QUE) and prove that the pairing between a quantum double group a...

Non-standard quantum groups $C_R [GL(n)]$ and $C_R [SL(n)]$ are constructed for a two parameter version of the Cremmer-Gervais $R$-matrix. An epimorphism is constructed from $C_R [GL(n)]$ onto the restricted dual $U_{\bar{R}}(\frak{gl}(n-1))$ associated to a related smaller $R$-matrix of the same form. A related result is proved concerning factoriz...

Let D be the factor of the enveloping algebra of a semisimple Lie algebra by its minimal primitive ideal with trival central character. We give a geometric description of the Chern character ch: K0(D)→HC0(D) and the state (of the maximal ideal m) s: K0(D)→K0(D/m) = in terms of the Euler characteristic χ:K0()→, where is the associated flag variety.

The primitive ideals of the quantum group C q [SL(n)] are classied in the case where q is a non-zero complex number which is not a root of unity. It is shown that the orbits in Prim C q [SL(n)] under the action of the character group H = (C ) n 1 are parameterized naturally by W W where W is the associated Weyl group. It is shown that there is a na...

The primitive ideals of the quantum group Cq[SL(n)] are classified in the case where q is a non-zero complex number which is not a root of unity. It is shown that the orbits in Prim Cq[SL(n)] under the action of the character group H ≅ (C*)n−1 are parameterized naturally by W × W where W is the associated Weyl group. It is shown that there is a nat...

The type-A Kleinian singularities are the surfaces of the form C2/G where G is a finite cyclic subgroup of SL2(C). The standard noncommutative analog is the fixed ring of the Weyl algebra A1(C) under the induced action of G. These fixed rings are, however, only part of a large family of algebras whose associated graded ring is the coordinate ring o...

The primitive ideals of the Hopf algebra C_q[SL(3)] are classified. In particular it is shown that the orbits in Prim C_q[SL(3)] under the action of the representation group H ≅C*×C* are parameterized naturally by W×W, where W is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals of C_q...

Let R be a Noetherian algebra over a field k. A formula is given for the Krull dimension of the ring Rk k(X) in terms of the heights of simple modules with large endomorphism rings.

A complete classification up to isomorphism is given of the fixed rings of the Weyl algebra with respect to the action of a finite group. The (Quillen) higher K-groups and (in most cases) the trace group or zeroth homology, are calculated for these rings.

Without Abstract

Without Abstract

Soit d un entier positif plus grand que 2, soit F un corps de caracteristique ne divisant pas d, et soit f une forme binaire diagonale de degre d. On note par Cf l'algebre de Clifford associee a f. On considere des images homomorphes de Cf et on montre qu'elles peuvent etre vues comme des produits croises sur l'anneau des coordonnees d'une courbe p...

A number of results are proved concerning the Quillen K-theory K*(S*G) of the skew group ring S*G, where S is a Noetherian ring and G is a finite group of automorphisms of S. Applications are given to the computation of K-groups of group algebras and of equivariant K-theory for affine varieties.

Let {s_i: 1 ≤ i < ∞} be a set of strictly positive integers. We produce an example of a ring R such that (a) R is a Noetherian domain, integral over its center, of (classical or Rentschler-Gabriel) Krull dimension one and (b) for each i, there exists an indecomposable, finitely generated, projective right R-module P_i such that P_i has uniform dime...

Let $R$ be an hereditary Noetherian prime ring, let $S$ be a "Dedekind closure" of $R$ and let $\mathcal{T}$ be the category of finitely generated $S$-torsion $R$-modules. It is shown that for all $i \geq 0$, there is an exact sequence $0 \rightarrow K_i(\mathcal{T}) \rightarrow K_i(R) \rightarrow K_i(S) \rightarrow 0$. If $i = 0$, or $R$ has finit...

For a hereditary noetherian prime ring R with classical quotient ring Q, various necessary and sufficient conditions are given for the polynomial ring R[X1,…, Xn] to be primitive when R itself is not primitive. It is shown that if R is a local hereditary noetherian prime ring, then R[X] is primitive if and only if Q[X] is primitive. Similarly, for...

Let X be an irreducible algebraic variety defined over a field k, let R be a sheaf of (noncommutative) noetherian k-algebras on X containing the sheaf of regular functions O and let R be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of R obtained from the...

Let X be an irreducible algebraic variety defined over a field k. let ℛ be a sheaf of (noncommutative) noetherian k-algebras on X containing the sheaf of regular functions 0 and let R be the ring of global sections. We show that under quite reasonable abstract hypotheses (concerning the existence of a faithfully flat overring of R obtained from the...

We investigate the relation between the Krull dimension of a ring K and of an extension R of K by considering modules of the form N⊗KR, where N is a simple or critical K-module.