Antisymmetric Infinitesimal Admissible Hom-bialgebras

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

Decompositions in categories of highest weight modules

Rota–Baxter operators on Witt and Virasoro algebras

The aim of this paper is to study some brackets defined on (τ,σ)-derivations satisfying quasi-Lie identities. Moreover, we provide examples of ( p , q ) ${(p,q)}$ -deformations of Witt and Virasoro algebras as well as 𝔰 ⁢ 𝔩 $\mathfrak{sl}$ (2) algebra. These constructions generalize the results obtained by Hartwig, Larsson and Silvestrov on σ-derivations, arising in connection with discretizations and deformations of algebras of vector fields.

ABSTRACTThe concept of -differential operators is a natural generalization of differential operators and difference operators. In this paper, we determine the -differential Lie algebraic structure on the Witt algebra and the Virasoro algebra for invertible . Then we consider several families of modules over the Virasoro algebra with explicit module actions and determine the -differential module structures on them.

In this paper, we establish the connection between the quantized W-algebra of and quantum parafermions of that a shifted product of the two quantum parafermions of generates the quantized W-algebra of

Following V. Drinfeld and G. Olshansky, we construct Manin triples such that g is different from Drinfeld’s doubles of a for several series of Lie superalgebras a which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization of suitable Etingof–Kazhdan’s results guarantee then the uniqueness of q-quantization of our Lie bialgebras. Our examples give solutions to the quantum Yang-Baxter equation in the cases when the classical YB equation has no solutions. To find explicit solutions is a separate (open) problem. It is also an open problem to list (à la Belavin-Drinfeld) all solutions of the classical YB equation for the Poisson superalgebras and the exceptional Lie superalgebra which has a Killing-like supersymmetric bilinear form but no Cartan matrix.

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.

Deformations of Lie algebras using σ-derivations

This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and the other for the closed string sector. Physical observables of quantum matrix models in the large N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient algebras of a larger Lie algebra. We will also discuss some properties of these Lie algebras not published elsewhere yet, and briefly review their relationship with well-known algebras like the Cuntz algebra, the Witt algebra and the Virasoro algebra. We will also review how the Yang–Mills theory, various low energy effective models of string theory, quantum gravity, string-bit models, and the quantum spin chain models can be formulated as quantum matrix models. Studying these algebras thus help us understand the common symmetry of these physical systems.

More on the classical symmetries of the closed bosonic p-branes

Introduction. Let G1 and G be connected semisimple algebraic groups defined over a field K of characteristic zero and assume that there is an isomorphism f of G1 onto G which is defined over R, the algebraic closure of K. If p: G -? GL(V) is an absolutely irreducible (finite-dimensional) representation of G defined over K, then p of is an absolutely irreducible representation of G1 defined over K. Satake [7, p. 230] has shown that there is a field K1 which is a finite extension of K, a (unique) central simple division algebra K# defined over K1, a finite-dimensional right vector space V1 over K#, and a K1-homomorphism pi: G1 -GL(V1/K#) (the group of all nonsingular K#-linear endomorphisms of V1) such that (p of)(g) = 01(p1(g)) for all g E G1 where 01 is a unique absolutely irreducible representation of End (V1/K#) (the algebra of all K#-linear endomorphisms of V1) onto End (V). In this paper we are interested in the case where K= K1 and where there are invariant forms on V and V1. More precisely, we state the following two problems. PROBLEM 1. Assume that K#=K and that there are invariant bilinear forms B on V and B1 on V1 which are defined over K. What is the relationship between these two forms over K? Of course, if B is alternating, so is B1 and both are determined by dim V=dim V1. Hence, we shall always take B and B1 to be symmetric. PROBLEM 2. Assume that K# is a nontrivial division algebra over K (i.e., K# AK) and that there is an invariant bilinear form B on V and an invariant ?hermitian form F (c = + 1 or 1) on V1 both of which are defined over K. What is the relationship between these two forms over K? We are especially interested in the case K= Q,, a p-adic field. (In a future paper, we shall discuss the case K=R.) Here, some simplifications are immediately available. In Problem 2, it can be shown [7, p. 232] that K# has an involution of the first kind; but over Q,, it is known that the only such division algebra is the quaternion division algebra. Furthermore, it is known that a hermitian form on a finite-dimensional vector space over a quaternion division algebra defined over Q, is determined only by the dimension of the vector space. Therefore, in Problem 2 we shall always take F to be skew-hermitian; in the case where K# is a quaterion division algebra, this means that the form B is symmetric [7, p. 233]. If W is a vector space defined over K and if S is a symmetric form on W which is also defined over K, then three invariants can be associated with the pair (W, S),

In this contribution we give results concerning the low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in the general tensor densities modules \(\mathcal {F}^\lambda \). More precisely, we provide the first and the second algebraic cohomology of the Witt and the Virasoro algebra with values in \(\mathcal {F}^\lambda \) for every \(\lambda \in \mathbb {C}\), and we give results for the third algebraic cohomology for certain values of \(\lambda \) in the case of the Witt algebra. The low-dimensional algebraic cohomology vanishes except for certain values of \(\lambda \) for which the dimension of the cohomology is one or two, depending on \(\lambda \). We consider algebraic cohomology, i.e. we do not put any continuity restrictions on our cochains and cocycles. Moreover, the given results are independent of any concrete realization of the Lie algebras under consideration, and thus, independent of any choice of topology.

We explain how the recently again discussed N-point Witt, Virasoro, and affine Lie algebras are genus zero examples of the multi-point versions of Krichever–Novikov type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. As example, explicit expressions for the three-point situation are given.

For finite-dimensional Lie algebras there is a tight connection between cohomology theory and deformations of the Lie algebra. This is not the case anymore if the Lie algebra to be deformed is infinite dimensional. Such Lie algebras might be formally rigid but nevertheless allow deformations which are even locally non-trivial. In joint work with Alice Fialowski the author constructed such geometric families for the formally rigid Witt algebra and current Lie algebras. These families are genus one (i.e. elliptic) Lie algebras of Krichever–Novikov type. In this contribution the results are reviewed. The families of algebras are given in explicit form. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics.