Witt Theorem Research Articles

Some quantum Lie algebras of typeDnpositive

A quantum Lie algebra is constructed within the positive part of the Drinfeld–Jimbo quantum group of type Dn. Our quantum Lie algebra structure includes a generalized antisymmetry property and a generalized Jacobi identity closely related to the braid equation. A generalized universal enveloping algebra of our quantum Lie algebra of type Dn positive is proved to be the Drinfeld–Jimbo quantum group of the same type. The existence of such a generalized Lie algebra is reduced to an integer programming problem. Moreover, when the integer programming problem is feasible we show, by means of the generalized Jacobi identity, that the Poincare–Birkhoff–Witt theorem (basis) is still true.

Un Theoreme de Poincaré–Birkhoff–Witt pour les Algebres de Leibniz

Poincaré–Birkhoff–Witt's theorem for Leibniz algebras. The dialgebras have been recently introduced by J. L. Loday. For any Leibniz algebra is defined an enveloping dialgebra which has a good universal property. In this work we give, in the category of dialgebras, a Poincaré–Birkhoff–Witt's theorem for Leibniz algebras: Let g be a Leibniz algebra, g Lie the associated universal Lie algebra and S(g Lie ) the symmetric algebra on the K-module g Lie . Let grUd(g) be the associated graded dialgebra of the enveloping dialgebra Ud(g) of g. Then, if g is free as K-module, there is an isomorphism of graded dialgebras S(g Lie ) ⊗ g = grUd(g).

A Poincaré–Birkhoff–Witt theorem for generalized Lie color algebras

A proof of the Poincaré–Birkhoff–Witt theorem is given for a class of generalized Lie algebras closely related to the Gurevich S-Lie algebras. As concrete examples, we construct the positive (negative) parts of the quantized universal enveloping algebras of type An and Mp,q,ε(n,K), which is a nonstandard quantum deformation of GL(n). In particular, we get, for both algebras, a unified proof of the Poincaré–Birkhoff–Witt theorem, and we show that they are genuine universal enveloping algebras of certain generalized Lie algebras.

Extension of Isometries in Finite-Dimensional Indefinite Scalar Product Spaces and Polar Decompositions

Witt's theorem on the extension of H-isometries to H-unitary matrices with respect to the scalar product generated by a self-adjoint nonsingular matrix H is studied in detail. All possible extensions are given, and their structure as a real analytic manifold is described. Analogous problems with respect to skew-symmetric scalar products are studied as well. The main motivation to study these problems, as well as the main applications of the results obtained, concerns polar decompositions in indefinite scalar product spaces. As another application, for given B all solutions of the matrix equation XA = B with H-unitary X and upper triangular A are described. Equations of this type are of vital importance in hyperbolic QR decompositions.

The pbw theorem, and generators and relations of the symmetric group

Let a group G act on a set Y. If vector values D(y,t) are given for all y∈ Y and some t ∈ G, then when does the equation admit a single‐valued solution F(y)? Necessary and sufficient conditions are given when G is a symmetric group and t is in a certain set of order two generators. A proof of the Poincaré ‐ Birkhoff ‐ Witt theorem is derived as an application.

Transitive extensions of imprimitive groups

We generalize the well-known Witt Theorem about the extensions of multiply transitive groups to imprimitive groups which are transitive on some t-tuples ( t ⩾ 2) having a special property. A nontrivial application is also given.

Poincaré–Birkhoff–Witt Theorem for Quadratic Algebras of Koszul Type

In this paper we prove a general Poincaré–Birkhoff–Witt theorem for quadratic Koszul algebras. The result is similar to that obtained by Polischuk and Positselsky, but the proof is entirely different and uses algebraic deformation theory. In particular, we obtain a new proof of the classical PBW-theorem for Lie algebras, based on deformation theory.

Schur Indices and Schur Groups

w x Schur groups were introduced in 8 as a certain kind of ‘‘smallest’’ groups admitting characters with nontrivial Schur index. Indeed every Ž . finite group G having an irreducible character x with m x / 1 involves such Schur groups. They occur as terminal steps by repeated application of Ž . the Brauer]Witt theorem over local fields of characteristic zero . Ž . There is a connection with the Schur subgroup S K of the Brauer Ž . Ž . group Br K of such a field K. This S K has been studied in detail by w x w x Yamada 12 and, quite recently, by Adem and Milgram 1 . From the character-theoretic point of view it appears to be of interest to give explicit Ž . Ž . generators for S K in terms of Schur groups see Section 6 . In the course w x of this investigation the authors noticed that there must be a gap in 8 Ž . Ž concerning the exceptional dyadic Schur groups yielding nontrivial 2-lo. cal index . It turned out that the definition of the Schur groups of type Ž . QD, q was too restrictive. The correct definition will be given in Section 5 below. We take the opportunity to discuss these ‘‘terminal Q -reduc2 Ž . tions’’ once again, and the class of 2-groups involved Theorem 4 and 5 . The Schur groups admit faithful irreducible characters which are induced from linear characters of cyclic normal subgroups. For the sake of clarity we begin our present work by making some general comments on Schur indices in group extensions. This should be compared with Turull w x 11 . Our approach is perhaps more elementary, and we just demonstrate Ž . w x how to use the results on stable Clifford theory 7 . More details may be w x found in the recent thesis 5 of the first-named author.

Witt's Extension Theorem for mod Four Valued Quadratic Forms

Witt's Extension Theorem for mod Four Valued Quadratic Forms

A PoIncare–BIrkhoff–Witt Theorem for Quantized Universal Enveloping Algebras of Type $A_N$

In this paper, we construct an explicit basis of the quantized universal enveloping algebra Uq(slN + 1(C)). Let A = (%-)i<jj<;v be a symmetrizable generalized Cartan matrix, and &(A) the Kac-Moody Lie algebra of A. Motivated by studies of YangBaxter equations, Jimbo [6] and Drinfeld [2, 3] introduced a Hopf algebra Uq(&(A)) with a nonzero complex parameter q. This Hopf algebra, which is also called [3] a quantum group, can be considered as a natural ^-analogue of the universal enveloping algebra U(&(A)) of &(A). For example, it is known that the representation theory of Uq(&(A)) is quite analogous to that of U(&(A)). See Lusztig [9] and Rosso [11]. The purpose of this paper is to show that, if &(A) is of type AN, and q 8 =£ 1, then Uq(&(A)) has a PoincareBirkhoff-Witt type basis. Let R be a commutative ring with 1. Denote by slN+l(R), the Lie algebra of (N + 1) x (N + 1) matrices over R of trace 0. It has the standard J^-basis consisting of the elements

Orbital stability of periodic solutions of autonomous systems with impulse effect

The orbital asymptotic stability of the periodic solutions of autonomous systems with an impulse effect is investigated. An analogue of the Andronov–Witt theorem is proved.

FILTRATIONS OF 2-ELEMENTARY FORMS AND INVOLUTIONS OF INTEGRAL SYMMETRIC AND SKEW-SYMMETRIC BILINEAR FORMS

In connection with the study of involutions of integral symmetric bilinear forms, for a finite quadratic (symmetric bilinear) 2-form there is defined on the annihilator of 2 a filtration of finite quadratic (symmetric bilinear) forms. The image of the group of automorphisms of the 2-form in the group of automorphisms of the filtration is determined, and an analogue of Witt's theorem is proved for these groups. Bibliography: 15 titles.

Witt's theorem for square-free lattices over dyadic discrete valuation rings

Witt's theorem for square-free lattices over dyadic discrete valuation rings

A generalized weight for linear codes and a Witt-MacWilliams theorem

Let F be a finite field, H a subgroup of F ∗ of index ν, and α 1,…, α ν coset representatives. For each n-tuple u = ( u 1,…, u n ) ϵ F n define W H ( u) = ( w 1( u),…, w ν ( u)), where w m ( u) = #{ u i : u i ϵ α m H}. An H-monomial map on F n is an automorphism of F n whose matrix with respect to the co-ordinate basis is of the form P · D, where P is a permutation matrix and D is a diagonal matrix with non-zero entries from H. Suppose C is an ( n, k) code over F (that is, a k-dimensional subspace of F n ) and that ϕ: C → F n is an injective homomorphism which preserves W H in the sense that W H ( ϕ(u)) = W H (u) for all u ϵ C. We prove that ϕ may be extended to an H-monomial map on F n . This generalization of a theorem of MacWilliams on the (Hamming) equivalence of codes may be considered an analogue of the Witt theorem of metric vector space theory.

Witt's Theorem for Modular Lattices

Witt's Theorem for Modular Lattices

Quadratic and hermitian forms in additive and abelian categories

During the last few years several papers concerned with the foundations of the theory of quadratic forms over arbitrary rings with involution have appeared. It is not necessary to give detailed references, in particular one thinks of the well known work of Bak [l], Bass [3], Karoubi, Knebusch [ll, 121, Ranicki, Vaserstein, and C. T. C. Wall. During the same period a number of problems quite similar to those occuring in the theory of quadratic forms were discussed, which, however, did not fit in the formalism developed so far. For example, one thinks of problems like the classification of pairs of forms, of sesquilinear forms, isometries , quadratic spaces with systems of subspaces, and also of quadratic forms over schemes, see e.g. [12, 13, 20, 22, 231. This situation called for a more general foundation of the theory of quadratic and hermitian forms. In this paper we try to give this foundation. Our basic object is an additive category &! together wit a duality functor *: & + A. In this situation one can define the most important notions of the theory of quadratic forms. Under suitable finiteness conditions one can prove a Krull-Schmidt theorem which is a sharpening of the classical Witt theorem. This result is basic for applications to the problems mentioned above. A preliminary version of this material is contained in [ 171. As just one application we discuss the classification of quadratic spaces with four subspaces. We hope that this disucssion will show clearly how one can solve a number of important classification problems of linear algebra. More applications can be found in [15, 16, 17, 22, 241. In the second part of the paper we discuss hermitian (not quadratic) forms in an abelian category. In an abelian category one has more structure, in particular one can introduce the notion of orthogonality. This allows one to introduce Grothendieck and Witt groups analogous to the G-groups in linear algebraic Ktheory which are obtained by factoring out exact sequences. As a basic result a Jordan-Holder theorem is proved for categories where all objects are of finite length. Using this theorem the computation of the Grothendieck group is 264

A Witt Theorem for Non-Defective Lattices

In [10], Witt laid the foundation for the study of quadratic forms over fields. Suppose Q is a quadratic form defined on a finite dimensional vector space V over a field of characteristic not equal to 2. Witt showed that non-zero vectors x and y in V satisfying Q(x) = Q(y) can be mapped into each other via an isometry of the vector space V. More generally, if τ : W ⟶ W’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

Unitary groups over semilocal domains. I

Let O be a semilocal domain that contains 2 as a unit. Let ∗ be an involution on O with the property that there exists a unit θ in O such that θ ∗ = −θ . In this paper, we study the structure of a lattice L with canonical splitting (see Section 2) over O and the generation theorems for the unitary group U( L) of the lattice and its commutator subgroup Ω(L). First, we prove the Witt theorem, i.e., every isometry from an orthogonal component of L into L can be lifted up to L, listed as Theorem 3.7. Then, we determine all the invariant sublattices of L (see Theorem 3.9). Let E( L) be the subgroup of U( L) generated by pure unitary transvections and pure quasi-transvections. We show that Ω(L) = E(L) and U( L) is generated by E( L) and the semideletions P H (α) (Listed as Theorems 4.4 and 4.5).

A non-Lie algebraic framework and its possible merits for symmetry descriptions

A nonassociative algebraic construction is introduced which bears a relation to a Lie algebra ℒ paralleling the relation between an associative enveloping algebra and ℒ. The key ingredient of this algebraic construction is the presence of two parameters which relate it to the enveloping algebra of ℒ. The analog of the Poincaré–Birkhoff–Witt theorem is proved for the new algebra. Possibilities of physical relevance are also considered. It is noted that, if fully developed, the mathematical framework suggested by this new algebra should be non-Lie. Subsequently, a certain scheme resulting from specific considerations connected with this (non-Lie) algebraic structure is found to bear striking resemblance to a recent phenomenological theory proposed for explaining CP violation by the K0 system. Some relevant speculations are also made in view of certain recent trends of thought in elementary particle physics. Finally, in an appendix, a Gell-Mann–Okubo-like mass formula for the new algebra is derived for an SU (3) octet.

Witt's theorem for noncommutative semilocal rings

Witt's theorem for noncommutative semilocal rings