Subalgebras of differential algebras with respect to new multiplications тема диссертации и автореферата по ВАК РФ 01.01.06, кандидат наук Бауыржан Каирбекович Сартаев

• Бауыржан Каирбекович Сартаев
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Бауыржан Каирбекович Сартаев

Contents

1 Introduction

1.1 General description of the work

1.1.1 The goal of research

1.1.2 General methodology of the research

1.1.3 Scientific novelty

1.1.4 Theoretical and practical significance

1.1.5 Publications

1.1.6 The structure and scope of the thesis

1.1.7 Acknowledgements

2 Differential Poisson algebras

2.1 Preliminaries on the special GD-algebras

2.2 Special GD-algerbas

2.3 Special identities of GD-algebras

2.4 On the Grobner basis of the Gelfand-Dorfman operad with special identities

3 Differential associative algebras

3.1 Preliminaries on the Novikov and noncommutative Novikov algebras

3.2 Basic notions on the differential associative algebra

3.3 Free noncommutative Novikov algebra

3.4 Embedding into differential algebras

3.5 Alternative proof of Theorem

4 Differential perm algebras

4.1 Preliminaries on the left symmetric algebras

4.2 Novikov dialgebras and differential perm algebras

4.3 Universal enveloping left-symmetric dialgebras

4.4 A linear basis of the free SLS-algebra

4.5 A criterion of speciality

5 Perm algebras under commutator and anti-commutator

5.1 On the commutator and anti-commutator

5.2 The results on the commutator and anti-commutator of perm algebra

5.3 Proof of theorems

5.3.1 Proof of Theorem

5.3.2 Proof of Theorem

5.3.3 Proof of Theorem

5.3.4 Proof of Theorem

5.3.5 Proof of Theorem

5.3.6 Proof of Theorem

5.4 The basis of universal enveloping perm algebras for metabelian Lie algebras

5.5 Proof of Theorem

6 Conclusion

Введение диссертации (часть автореферата) на тему «Subalgebras of differential algebras with respect to new multiplications»

1. Introduction

Varieties of algebras are one of the fundamental concepts of abstract algebra. In many cases, a variety is defined as a class of algebras with a fixed set of operations satisfying a fixed family of universal axioms (identities). For instance, the well-known variety of associative algebras is a collection of vector spaces with one bilinear operation on vectors which satisfies associativity identity. An important result in the study of varieties of algebras is the Birkhoff Theorem which states that every class of algebras closed under Cartesian products, subalgebras, and homomorphic images is a variety, and vice versa. The study of varieties of algebras involves the methods of universal algebra and category theory, which is related to the general properties of algebraic structures.

The notion of a functor is a basic tool in the category theory that maps objects and morphisms of one category into objects and morphisms of another category. In particular, functors can be used to transform algebraic structures of one variety into algebras of another variety. An example of a such functor is a forgetful functor that maps the category of algebras into the category of vector spaces by forgetting the multiplication operation. Such a functor is essentially used in the study of L-varieties [32].

Also, functors can be used for studying relationships between different algebraic structures. Two different varieties of algebras Var A and Var B can be connected through some functor

f : Var A ^ Var B

that changes the operation(s) of Var A to get the operation(s) of Var B. An example of a such functor is the commutator or the anticommutator which is defined on algebras with one binary multiplication (see [48] for other known examples).

One of the main problems related to the study of operation-transforming func-

tors between varieties is to determine a set of identities that hold on the resulting systems. An identity that holds of f (Var A) but does not hold on the entire Var B is called special. The problem of finding special identities is solved or largely answered for some varieties, but it is still open for others. A classic illustration of this problem is related to Var A = As of associative algebras.

Namely, consider the functor (—) : As ^ M from As to the variety M of all (nonassociative, or magmatic) algebras with one multiplication, such that each A E As transforms to the commutator algebra A(-): the same space A with new operation [a, b] = ab — ba, a,b E A. Then the classical Poincare-Birkhoff-Witt (PBW) Theorem states that ass special identities are corollaries of anti-commutativity [a, b] + [b, a] = 0 and Jacobi identity [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0.

However, for the anti-commutator functor (+) : As ^ M transforming A E As to the algebra A(+) with new operation {a, b} = ab + ba there is no complete knowledge on special identities (see [47]). In fact, the operation {•, •} satisfies commutativity and Jordan identity (of degree 4), but there is also an independent Glennie's identity (of degree 8), and the complete list of special identities is still unknown.

In order to find special identities for a pair Var A and Var B relative to a functor f we consider the multilinear part of the free algebra Var A(X) in Var A generated by a countable set of variables X = {x1,x2,... }. This multilinear part carries the structure of a symmetric operad [46]. Note that a notion equivalent to symmetric operads was introduced in [1]. In a formal way, an operad can be defined in terms of generators and relations. At the level of operads, generators are algebraic operations and defining relations correspond to multilinear identities on Var A(X).

Let us state the formal definition of an operad and explain the relation between a variety of algebras and the corresponding operad. Let Sn stand for the symmetric group of the set {1,... ,n}. A symmetric operad P is a collection of Sn-modules P(n), n > 1, equipped with linear composition maps

7m,..,nm : P(m) 0 P(ni) 0 ... 0 P(nm) ^ P(ni + ... + nm)

which satisfies the natural associativity condition. The space P(1) should contain an element id that acts as an identity relative to the compositions. Finally,

the compositions have to be equivariant with respect to the symmetric group actions. The composition maps can be demonstrated on the following example. Assume P(n) is spanned by planar binary trees with n leaves that labelled by numbers from 1 to n. In this case, the composition of f £ P(m) with f £ P(n), i = 1,... ,m, means that each f is attached to f at the corresponding leave i, and the numbers of labelled leaves of f are shifted in a proper way. The construction is explicitly described in [10].

Given a variety Var A of algebras (with binary operations), each tree with n leaves represents a multilinear monomial in x\,... ,xn of the algebra Var A(X), the numbers of leaves correspond to the variables x\,... ,xn £ X. A composition can be understood as the substitution of monomials, the action of Sn permutes the variables. For example, if f = x2(xix3), f1 = xi, f2 = (x3x^x2, f3 = x2x1

then Yl33,2(f,f1,f2,f3) = ((x4x2)x3)(x1 (x6x5)).

The operad Com-As defining the variety of commutative-associative algebras is generated by single operation ^ = x1x2 = = x2x1, and the only defining relation of Com-As is the associativity identity:

Y221(M,M,1) + yi22(m, 1,M).

We will use the same notation for a variety of algebras and for the corresponding operad.

Given a vector space V, one may also define an operad also denoted V with V(n) = Hom (V0n, V).

A morphism of operads ^ : O ^ P is a collection of Sn-linear maps ^(n) : O(n) ^ P(n), n > 1, preserving the compositions and identity. The structure of a Var A-algebra on a vector space V is the same as a morphism of operads Var A ^ V. Therefore, an operation-changing functor f from Var A to Var B is determined by a morphism of operads ^ : Var B ^ Var A so that f (V) is the composition of ^ and the Var A-algebra structure on V:

Var B ^ Var A ^ V.

Another problem of studying an operation-changing functor f for a pair of varieties Var A and Var B is to define which algebras of Var B are embeddable into f (Var A). Such algebras are called special. It is well known that if Var A =

As is the variety of associative algebras and Var B = Lie is the variety of Lie algebras then, relative to the functor (—), all Lie algebras are special. In the case when f is the anti-commutator functor (+) : As ^ Jord, where Jord is the variety of Jordan algebras, there exist non-special (exceptional) algebras. One of the classical results in this direction is that any two-generated Jordan algebra is special, but it is well-known that homomorphic image of a special Jordan algebra may be exceptional.

A variety Var of (binary) algebras may be enriched with a derivation. A derivation d on an algebra A is a linear mapping d : A ^ A satisfying the Leibniz rule

d(ab) = d(a)b + ad(b), a,b E A.

An algebra A with derivation d is called a differential algebra. The class of differential algebras in a variety Var also forms a variety denoted by Var(d\ This variety is governed by an appropriate operad obtained from Var by adding one more generator d (unary operation) and one more defining relation

72(d M) = Yi, 1 (M d, id) + Y i, 1 (Mid, d)

which reflects the Leibniz rule.

Let us define a functor f from a variety Var(d) to a variety of algebras

with two operations > and -<. The functor f changes the operations of multiplication in a differential algebra from Var(d to the new operations as follows:

x 1 > x2 = d(x 1 )x2, x 1 -< x2 = x 1d(x2).

Namely, the variety Var(>,<S) is generated by all algebras of the form (A, >, -<) for all (A, d) in Var(d. Using the white Manin product for operads, the complete list of identities that hold on the multiplications > and -< in all algebras of the variety Var(>,<S) was found in [35].

In this work we mainly consider the described problem of finding special identities and resolve the question of speciality for some pairs of varieties related with operation-changing functors

f : Var(d)

^ Var(<S) (1.1)

and

f : Var(d) ^ Var(y,<\ (1.2)

The functor (1.1) we consider for the cases when Var is the variety of Poisson algebras or perm algebras. The functor (1.2) we consider for both associative and perm algebras.

An algebra obtained from a differential algebra (A, d) under new operation a -< b = ad(b) or/and with an additional one a y b = d(a)b is said to be a derived algebra. In the sequel we will omit the functor f and use the term "derived Var-algebra" for a differential algebra A £ Var(d) under these new operations.

The problem considered in Chapter 2 is to describe the class of algebras that are embeddable into differential Poisson algebras under the following new operations:

x o y = x ^ y = xd(y), [x,y} = {x,y}.

Under these products, a differential Poisson algebra turns into a Gelfand-Dorfman (shortly, GD-) algebra. A Gelfand-Dorfman algebra is called special if it can be embedded into some differential Poisson algebra. Non-special GD-algebras exist: the examples were found implicitly in [35] and explicitly in [34]. Note that all these examples are of dimension three. This is a reason to study the class of special Gelfand-Dorfman algebras and special identities of Gelfand-Dorfman algebras.

The definition of a GD-algebra emerged in the paper [26] as a tool for constructing Hamiltonian operators in formal calculus of variations. Later it was shown [64] that GD-algebras are in one-to-one correspondence with quadratic Lie conformal algebras playing an important role in the theory of vertex operators. The class of GD-algebras is governed by a binary quadratic operad (in the sense of [29], see also [10]) denoted GD. As it was shown in [35], the Koszul dual operad GD! corresponds to the class of differential Novikov-Poisson algebras introduced in [8]. The latter algebras were shown to play an important role in the combinatorics of derived operations on non-associative algebras [35].

In chapter 3, we describe the derived algebras of associative algebras with a derivation relative to two new products

a -< b = ad(b), a y b = d(a)b. The motivation for this chapter comes from the well-known fact that an associative

and commutative algebra A with a derivation d : A ^ A turns into a Novikov algebra when equipped with the new bilinear operation

a o b = ad(b), a,b E A. (1.3)

The study of the structure theory of Novikov algebras was initiated by [67], where it was shown that a finite-dimensional simple Novikov algebra over an algebraically closed field of zero characteristic is 1-dimensional. Further development of the structure and representation theory of Novikov algebras was obtained in [50, 51, 65, 66].

Significant progress in the combinatorial study of Novikov algebras was achieved in [24], where a monomial basis of the free Novikov algebra was found. It turned out that the free Novikov algebra Nov(X) generated by a set X embeds into the free commutative differential algebra ComDer(X; d) relative to the operation (1.3). The latter may be considered as the polynomial algebra in the variables X M = X U X' U X" U ... with respect to obvious derivation. As a result, it follows that the defining identities of Novikov algebras exhaust independent special identities relative to the operation-changing functor described by (1.3). Recent advances in the combinatorial theory of Novikov algebras include the study of algebraic dependence [18], nilpotence and solvability [58, 68].

In [7], the Grobner-Shirshov bases theory for Novikov algebras was developed. The constructions and proofs of [7] essentially depend on the results of [24]. It was proved in [7] that every Novikov algebra may be embedded into an appropriate commutative differential algebra relative to the operation (1.3). As a result, every Novikov algebra is special relative to the functor (1.3).

In the dissertation, we consider the same problems in non-commutative setting: define non-commutative Novikov algebras and prove that they are all special, i.e., can be embedded into associative algebras with a derivation.

In Chapter 4 and Chapter 5, we consider derived algebras of perm algebras under multiplication (1.3), commutator, and anti-commutator. More precisely, perm algebra under the operation (1.3) turns into a left-symmetric algebra. The class of left-symmetric algebras (also known as pre-Lie algebras) initially appeared in geometry and deformation theory [62], [40], [28]. It is worth noting that the variety of left-symmetric algebras contains the variety of Novikov algebras. We will find all special identities in these settings and study the problem of speciality.

In addition, we study perm algebras under (anti-)commutator. Perm algebras under commutator and anti-commutator give metabelian Lie algebras and Jordan algebras with 2 additional identities, respectively. In contrast to the general associative algebra case, all special identities for anti-commutator on perm algebras are found.

A variety of metabelian Lie algebras is defined by the additional identity

[[a,b], [c, d]] = 0. (1.4)

This variety has received a lot of attention and was considered in [15], [25], [57], [41], [42], [55]. A basis of the free metabelian Lie algebras was given in [2]. Using this basis, we obtain one of the main results of the dissertation, an analogue of the Poincare-Birkhoff-Witt Theorem for the embedding of metabelian Lie algebras into perm algebras.