Chapter VI. Multilinear Algebra

We will consider real or complex vector spaces (K = R or C), E 1, E 2, ..., E r of dimensions n 1, n 2, ..., n p. In this first part, we will only recall some results, omitting all proofs, only to specify our definitions, our notations and our methods. We will assume that the reader is already somewhat familiar with tensor algebras.

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincare-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Grobner-ready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Grobner bases for commutative and non-commutative algebras. It was T. Stokes who eighteen years ago introduced Grobner left bases (GLB) and Grobner left ideal bases (GLIB) for left ideals in Grassmann algebras, with the GLIB bases solving an ideal membership problem. Thus, a natural question is to first seek Grobner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levandovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Grobner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise this way. Examples of G-algebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided ideal, include Grassmann and Clifford algebras. After recalling basic concepts behind the theory of commutative Grobner bases, a review of the Grobner bases in PBW algebras, G-,and GR-algebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. GLB and GLIB bases will also be computed.

The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called “classical separation of variables theorem” for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.

Let Γ be a finite group acting faithfully and linearly on a real vector space V . Let T(V^∗) (S(V^∗)) be the tensor (symmetric) algebra associated to V^∗ which has a natural Γ action. We study generalized quadratic relations on the tensor algebra T(V^∗) ⋊ Γ. We prove that the quotient algebras of T(V^∗) ⋊ Γ by such relations satisfy the Poincare-Birkhoff-Witt (PBW) property. Such quotient algebras can be viewed as quantizations of linear or constant Poisson structures on T(V^∗) ⋊ Γ, and are natural generalizations of symplectic reflection algebras. We study these algebras via various examples and their related cohomologies.

Bilinear transformations and bilinear forms are introduced and studied. Their matrix representation, and especially the representation of symmetric bilinear forms, is presented. Orthogonality relative to a bilinear form is considered. These notions are then used to define the tensor product of vector spaces, and the properties of the tensor product are considered in detail.

This advanced textbook on linear algebra and geometry covers a wide range of classical and modern topics. Differing from existing textbooks in approach, the work illustrates the many-sided applications and connections of linear algebra with functional analysis, quantum mechanics and algebraic and differential geometry. The subjects covered in some

We study definable sets, groups, and fields in the theory of infinite-dimensional vector spaces over an algebraically closed field of any fixed characteristic different from 2 equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define a notion of dimension of a definable set, which enjoys many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all definable groups are (algebraic-by-abelian)-by-algebraic. We conclude that every definable field is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component.We also prove analogous results working over real closed fields.

A symmetric Lie algebra is defined, following J. Dixmier, to be a Lie algebra 9 with a decomposition 9 = I © p such that £ is a subalgebra of 9. [t.pi c P and [p,p ) C f.A definition of Cartan subspace of a symmetric Lie algebra is given, and a theory is presented which parallels the standard theory of Cartan subalgebras of Lie algebras, and which generalizes the classical results for real and complex semisimple symmetric Lie algebras.Introduction.In this paper, we first develop a theory of Cartan subspaces of symmetric Lie algebras over infinite fields of any characteristic.Then we specialize to characteristic zero, and finally to reductive Lie algebras over characteristic zero.The results here are suggested by those in [1, §1.13], [4, §1.1] and [5, §2.5 and Exercises 13-15 at the end of Part 2].We are indebted to D. Haile for certain results in § §4 and 5.1. Weight theory for nil sets.In this section, we sketch a theory of weights in a slightly more general setting than the standard one.Throughout this section, 9 is a Lie algebra over a field k and it: 9 -► End V is a representation of 9 on a vector space V. (Vector spaces, Lie algebras and modules shall be finite dimensional throughout this paper.)Whenever 9 is explicitly regarded as a 9-module, we use the adjoint representation.Definition.If A is a subset of 9 and X is a function from A into k, V¿ is the set of x E 9 such that some power of it(a) -X(a) annihilates x for aU a EA.If A = {a}, write K*(a) for V% .The functions X such that V% ± 0 are weights, and any nonzero element of V¿ is called a weight vector for X.The set of weights will be denoted by A(F) or &A(V).Definition.A subset A of 9 is nil if A E 9° .(According to our convention, 77 is the adjoint representation here.)

Algebraic Studies of Symmetric Operators By Zhiqin Shi Dissertation Director: Professor William Keigher There was an old problem of G. C. Rota regarding the classification of all linear operators on associative algebras that satisfy algebraic identities. We only know very few of such operators at the beginning, for example, the derivative operator, average operator, difference operator and Rota-Baxter operator. Recently L. Guo, W. Sit and R. Zhang revisited Rota’s problem in a paper by concentrating on two classes of operators: differential type operators and Rota-Baxter type operators. One of the Rota-Baxter type operators they found is the symmetric Rota-Baxter operator which symmetrizes the RotaBaxter operator. In this dissertation, we initiate a systematic study of the symmetric Rota-Baxter operator, extending the previous works on the original Rota-Baxter operator. After giving basic properties and examples, we construct free symmetric Rota-Baxter algebras on an algebra and on a set by bracketed words and rooted trees separately. We then use the free symmetric Rota-Baxter algebra to obtain an extension of the well known dendriform algebra and its free objects. Finally, we extend our study to differential algebras. We construct the free symmetric differential Rota-Baxter algebra based on the previous free symmetric Rota-Baxter algebra on a set and the free symmetric differential algebra.

The symplectic vector spaceE of theq andp's of classical mechanics allows a basis free definition of the Poisson bracket in the symmetric algebra overE. Thus the symmetric algebra overE becomes a Lie algebra, which can be compared with the quantum mechanical Weyl algebra with its commutator Lie structure. The universality of the Weyl algebra is used to study the well-known ‘classical’ Moyal realisation of the Weyl algebra in the symmetric algebra. Quantisations are defined as linear mappings of the underlying vector spaces of the two algebras. It is shown that the classical Lie algebra is −2 graded, whereas the quantum Lie algebra is not. This proves that they are not isomorphic, and hence there is no Dirac quantisation.

This paper gives APL models of multilinear forms, tensors, symmetric and antisymmetric tensors, change of basis formula, thus providing a coherent treatment of symmetric and exterior algebras. Properties of determinants are derived from exterior algebra identities, and formal computation of coefficients of the characteristic polynomial of a square matrix follows as a natural consequence.

This chapter is devoted to the computation of Hochschild and cyclic homologies of some particular types of algebras: tensor algebras, symmetric algebras, universal enveloping algebras of Lie algebras and, finally, smooth algebras, on which we put some emphasis. One more important example, the case of group algebras, will be treated later, in Sect. 7.4. It is also shown in this chapter that Hochschild and cyclic homology are related to many other theories such as the homology of Lie algebras, André-Quillen homology of commutative algebras, and Deligne cohomology.KeywordsExact SequenceSpectral SequenceCommutative AlgebraPolynomial AlgebraFree ResolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The almost split triangles for perfect complexes over gentle algebras

It is a standard and very useful technique to approximate geometric objects and maps by linear objects and maps. A principal (trivial) example in basic analysis is approximating an open subspace U of \(\mathbb{K}^{n}\) at a point by the linear space \(\mathbb{K}^{n}\) itself by visualizing \(\mathbb{K}^{n}\) as a tangent space to U at the given point. Another (less trivial) example is the derivative of a differentiable map f on U in a point that is the best approximation of f nearby this point by a linear map. In the first section we extend this technique to premanifolds and morphisms of premanifolds. Thus we first define the tangent space of a premanifold at a point, which will be a \(\mathbb{K}\)-vector space that we visualize as the ”best linear approximation“ of the premanifold at the given point. Then we define the derivative in a point x of a morphism f. This will be a linear map from tangent space at x to the tangent space at \(f(x)\).

We start from a basic version of the Hahn-Banach theorem, of which we provide a proof based on Tychonoff’s theorem on the product of compact intervals. Then, in the first section, we establish conditions ensuring the existence of affine functions lying between a convex function and a concave one in the setting of vector spaces — this directly leads to the theorems of Hahn-Banach, Mazur-Orlicz and Fenchel. In the second section, we caracterize those topological vector spaces for which certain convex functions are continuous — this is connected to the uniform boundedness theorem of Banach-Steinhaus and to the closed graph and open mapping theorems of Banach. Combining both types of results readily yields topological versions of the theorems of the first section.KeywordsConvex FunctionTopological Vector SpaceBasic TheoremClosed GraphApply TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.