BASES FOR QUANTUM ALGEBRAS AND SKEW POINCARE-BIRKHOFF-WITT EXTENSIONS

BASES PARA ALGEBRAS CUANTICAS Y EXTENSIONES TORCIDAS DE POINCARE-BIRKHOFF-WITT

Armando Reyes¹, Héctor Suárez²

¹ Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia.
² Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia.
Armando Reyes: mareyesv@unal.edu.co

(Recibido: Septiembre/2016. Aceptado: Diciembre/2016)

Abstract

Considering quantum algebras and skew Poincaré-Birkhoff-Witt (PBW for short) extensions defined by a ring and a set of variables with relations between them, we are interesting in finding a criteria and some algorithms which allow us to decide whether an algebraic structure, defined by variables and relations between them, can be expressed as a skew PBW extension, so that the base of the structure is determined. Finally, we illustrate our treatment with examples concerning quantum physics.

Resumen

Para las álgebras cuánticas y las extensiones torcidas de Poincare-Birkhoff-Witt definidas por un anillo y un conjunto de variables con relaciones entre ellas, estamos interesados en establecer un criterio y algunos algoritmos que nos permitan decidir si una estructura algebraica, definida en términos de generadores y relaciones, puede expresarse como una extensión torcida de Poincare-Birkhoff-Witt, de manera que se determine la base de la misma. Ilustramos nuestro tratamiento con diversas algebras de la física cuántica.

Palabras clave: Álgebras cuánticas, extensiones torcidas de Poincare-Birkhoff-Witt, lema del diamante.

Introduction

Historically, the importance of quantum algebras has been considered for several authors in the context of quantum mechanics, see [1] and [2]. For instance, in [3] it was presented a purely algebraic formulation of quantum mechanics which does not require the specification of a space of state vectors; rather, the required vector spaces can be identified as substructures in the algebra of dynamical variables (suitably extended for bosonic systems). As we can see, this formulation of quantum mechanics captures the undivided wholeness characteristic of quantum phenomena, and provides insight into their characteristic nonseparability and nonlocality. In fact, and like the authors say in [3], "the formalism we present fulfils Dirac's aim of working with the algebra of quantum mechanics alone. Furthermore, this approach addresses Dirac's interpretational difficulty, since it can be interpreted in terms of a "process" approach to quantum theory".

Now, from a philosophical point of view, it is very important the new relationships between physics and mathematics that emerge with Heisenberg's discovery of matrix mechanics and its development in the work of Born, Jordan, and Heisenberg himself. Precisely, this is the Einstein's view of "the Heisenberg method", as "a purely algebraic method of description of nature". In [4], chapter 4, it is examined the shift from geometry to algebra in quantum mechanics as a reversal of the philosophy that governed classical mechanics by grounding it mathematically in the geometrical description of the behavior of physical objects in space and time (Heisenberg's matrix mechanics abandons any attempts to develop this type of description and instead offers essentially algebraic machinery for predicting the outcomes of experiments observed in measuring instruments).

One of the fundamental objects in quantum theory is the Heisenberg algebra (see [5] for a detailed exposition ofthis quantum algebra). This algebra and its generalizations - deformations -have recently become of interest in both theoretical physics and mathematics, where it is regarded as a fundamental object and as a suitable model for checking various physical and mathematical ideas and constructions (c.f. [6-15], and others). For example, in [15] it is discussed representations of the Heisenberg relation in various mathematical structures; in [12], it is investigated the structure of two-sided ideals - a key concept in noncommutative algebra - in the q-deformed Heisenberg algebras and the relationships of this algebra with the quantum plane, and its realizations are of primary importance to studying the dynamics of a q-deformed quantum system (see [11] for an exposition of the q-deformed Heisenberg algebra and its relation with the origin of q-calculus).

Actually, and following [14], "algebraic methods have long been applied to the solution of a large number of quantum physical systems. In the last decades, quantum algebras appeared in the framework of quantum integrable one-dimensional models and have ever since been applied to many physical phenomena [... ] It was found that it could be generalized leading to the concept of deformed Heisenberg algebras [16], that have been used in many areas, as nuclear physics, condensed matter, atomic physics, etc". Indeed, the algebraic approach in theoretical physics has been also considered in a possible reconciliation of the quantum mechanics with general relativity theory, where the gravity does not need to be quantized [17].

With this in mind, several families of algebras have been defined with the purpose of studying mathematical and physical properties of different algebraic systems. One of them are the skew Poincare-Birkhoff-Witt extensions (PBW for short) introduced in [18]. These extensions have been studied in several papers ([18-28], and others), and the PhD Thesis [29], where the first author studied ring and module theoretical properties of these algebras.

The paper is organized as follows. Section 1 contains the criteria and algorithms of our treatment. Section 2 is dedicated to definition and some properties of skew PBW extensions. Section 3 presents two examples of quantum algebras which illustrate the results established in Section 1 (other examples can be found in [29]). Finally, we present some conclusions about this topic and a future work.

1. Diamond Lemma and PBW Bases

Bergman's Diamond Lemma [37] provides a general method to prove that certain sets are bases of algebras which are defined in terms of generators and relations. For instance, the Poincare-Birkhoff-Witt theorem, which appeared at first for universal enveloping algebras of finite dimensional Lie algebras (see [30] for a detailed treatment) can be derived from it. PBW theorems have been considered several classes of commutative and noncommutative algebras (see [33-36], and others). With this in mind, and since skew PBW extensions are defined by a ring and a set of variables with relations between them (Definition 2.1), in this section we establish a criteria and some algorithms which decide whether a given ring with some variables and relations can be expressed as a skew PBW extension. This answer is obtained following the original ideas presented in [37] and the treatment developed in [29].

Definition 1.1. (i) Let X be a non-empty set and denote by 〈X〉 and R〈X〉 the free monoid on X and the free associative R-ring on X, respectively. A subset Q ⊆〈X〉×R〈X〉 is called a reduction system for R〈X〉. An element σ =(Wσ,fσ) ∈ Q has components Wσ a word in 〈X〉 and fσ a polynomial in R〈X〉. Note that every reduction system for R〈X〉 defines a factor ring A = R〈X〉/IQ, with IQ the two-sided ideal of R〈X〉 generated by the polynomials Wσ − fσ, with σ ∈ Q.

(ii) If σ is an element of a reduction system Q and A, B ∈〈X〉, the R-linear endomorphism rAσB : R〈X〉→ R〈X〉, which fixes all elements in the basis 〈X〉 different from AWσB and sends this particular element to AfσB is called a reduction for Q. If r is a reduction and f ∈ R〈X〉, then f and r(f) represent the same element in the R-ring R〈X〉/I_Q. Thus, reductions may be viewed as rewriting rules in this factor ring.

(iii) A reduction rAσB acts trivially on an element f ∈ R〈X〉 if rAσB(f)= f. An element f ∈ R〈X〉 is said to be irreducible under Q if all reductions act trivially on f. Note that the set R〈X〉_irr of all irreducible elements of R〈X〉 under Q is a left submodule of R〈X〉.

(iv) Let f be an element of R〈X〉. We say that f reduces to g ∈ R〈X〉, if there is a finite sequence r1,...,rn of reductions such that g =(rn ··· r1)(f). We will write f →Q g. A finite sequence of reductions r1,...,rn is said to be final on f, if (rn ··· r1)(f) ∈ R〈X〉_irr.

(v) An element f ∈ R〈X〉 is said to be reduction-finite, if for every infinite sequence r1,r2,... of reductions there exists some positive integer m such that ri acts trivially on the element (r_i−1 ... r₁)(f), for every i > m. If f is reduction-finite, then any maximal sequence of reductions r₁,...,rn such that ri acts non-trivially on the element (r_i−1 ··· r₁)(f), for 1 ≤ i ≤ n, will be finite. Thus, every reduction-finite element reduces to an irreducible element. We remark that the set of all reduction-finite elements of R〈X〉 is a left submodule of R〈X〉.

(vi) An element f ∈ R〈X〉 is said to be reduction-unique ifit is reduction-finite and if its images under all final sequences of reductions coincide. This value is denoted by r_Q(f).

Proof. (i) Consider f, g ∈ R〈X〉un,λ ∈ R. We know that λf + g is reduction-finite. Let r₁,...,r_m be a sequence of reductions (note that it is final on this element), and r := r_m ··· r₁ for the composition. Using that f is reduction-unique, there is a finite composition of reductions r' such that (r'r)(f)= r_Q(f), andin a similar way, a composition of reductions r'' such that . Hence, the expression r(λf + g) is uniquely determined, and λf + g is reduction-unique. In fact, , and therefore (i) is proved.

(ii) From (i) we know that fgh is reduction-unique. Consider . The idea is to show that fr(g)h is reduction-unique and r_Q(fr(g)h)= r_Q(fgh). Note that if f, g, h are terms A, B, C, then r_ADσEC(ABC)= Ar_DσE(B)C, that is, Ar_DσE(B)C is reduction-unique with r_Q(ABC)= r_Q(Ar_DσE(B)C).

Now, more generally, where the indices i, j, k run over finite sets, with λ_i,µ_j,ρk, and where A_i,B_j,C_k are terms such that A_iB_jC_k is reduction unique for every i, j, k. In this way, . Finally, since is reduction-finite for every i, j, k, and we have is reduction-unique and .

Proposition 1.3 ([29], Proposition 3.1.3). If every element f G R(X) is reduction-finite under a reduction system Q, and Iq is the ideal of R(X) generated by the set (W_σ - f_σ | σ ε Q} then R(X) = R(X)_irr ⊕ I_Q if and only if every element of R(X) is reduction-unique.

Proof. Suppose that . Note that if are elements for which f reduces to g and g', then , that is, f is reduction-unique. Conversely, if every element of R(X) is reduction-unique under Q, then tq : R(X) → R(X)_irr is a R-linear projection. Consider f G ker(rQ), that is, rQ(f) = 0. Then f G Iq, whence the ker(rQ) C Iq, but in fact, ker(rQ) contains Iq: for every a G Q,A,B G (X), we have from Proposition 1.2, when r = r_1σ1.

Under the previous assumptions, A = R(X)/I_Q may be identified with the left free R-module R(X)irr with R-module structure given by the multiplication f * g = r_Q(fg).

Definition 1.4. An overlap ambiguity for Q is a 5-tuple of the form such that Wσ- = AB and W_T = BC. This ambiguity is solvable if there exist compositions of reductions r, r' such that r(f_σC) = r'(Af_{T τ}). Similarly, a 5-tuple (σ, τ, A, B, C) with σ ≠ τ is called an inclusion ambiguity if W_τ = B and W_σ = ABC. This ambiguity is solvable if there are compositions of reductions r, r' such that r(Af_τB) = r'(f_σ).

Definition 1.5. A partial monomial order ≤ on (X) is said to be compatible with Q if f_σ is a linear combination of terms M with M < Wσ, for all σ ∈ Q.

Proposition 1.6 ([29], Proposition 3.1.6). If ≤ is a monomial partial order on (X) satisfying the descending chain condition and compatible with a reduction system Q, then every element f ∈ R(X) is reduction-finite. In particular, every element of R(X) reduces under Q to an irreducible element.

Definition 1.7. An overlap ambiguity (σ, τ, A,B,C) is said to be resolvable relative to < if f_oC - Af_T ∈ Y_ABC. An inclusion ambiguity (σ, τ, A,B,C) is said to be resolvable relative to < if Af_τC - fσ ∈ Y_ABC.

If r is a finite composition of reductions, and f belongs to V_M, then f - r(f) ∈ Y_M. Hence, f ∈ Y_M if and only if r(f) ∈ Y_M ([19], Proposition 3.1.8).

Proposition 1.8 (Bergman's Diamond Lemma [37]; [29], Theorem 3.21). Let Q be a reduction system for the free associative R-ring R(X), and let ≤ be a monomial partial order on (X), compatible with Q and satisfying the descending chain condition. The following conditions are equivalent: (i) all ambiguities of Q are resolvable; (ii) all ambiguities of Q are resolvable relative to ≤; (iii) all elements of R(X) are reduction-unique under Q; (iv) R(X) = R(X)_irr ⊕ I_Q.

1.1. Algorithms

Throughout this section we will consider the lexicographical degree order :≤_deglex to be defined on the variables x₁,... ,x_n. For more details about these orders, see [18], section 3.

Definition 1.9. A reduction system Q for the free associative R-ring R(x₁,..., x_n) is said to be a ≤_deglex-skew reduction system if the following conditions hold: (i) ; (ii) for every , where c_i;j ∈ R \ {0} and p_ji ∈ R(x₁,..., x_n); (iii) for each j > i, lm . We will denote (Q, ≤_degiex) this type of reduction systems.

Note that if , we consider its Newton diagram as . In this way, by Proposition 1.6 every element f ∈ R(x₁,...,x_n) reduces under Q to an irreducible element. Let Iq be the two-sided ideal of R(x₁,... ,x_n) generated by W_ji - f_ji, for 1 ≤ i < j ≤ n. If x_i + Iq is also represented by x_i, for each 1 < i < n, then we call standard terms in A. Proposition 1.11 below shows that any polynomial reduces under Q to some standard polynomial and hence standard terms in A generate this algebra as a left free R-module.

Proposition 1.10 (29], Lemma 3.2.2). If (Q, ≤_deglex) is a skew reduction system, then the set R(x₁,..., x_n)_irr is the left submodule of R(x₁,... ,x_n) consisting of all standard polynomials f ∈ R(X₁,. . . ,X_n).

Proof. It is clear that every standard term is irreducible. Now, let us see that if a monomial M = λx_j1 ... x_js is not standard, then some reduction will act non-trivially on it. If s < 2 the monomial is clearly standard. This is also true if j_k ≤ j_k+₁, for every 1 ≤ k ≤ s - 1. Let s ≥ 2. There exists k such that j_k ≤ j_k+1 and M = Cx_jx_iB = CW_jiB where j = j_k, i = j_k+1 and where C and B are terms. Then CW_jiB -q Cf_jiB acts non trivially on M.

Proof. It follows from Proposition 1.10 and Proposition 1.6.

Next, we present an algorithm to reduce any polynomial in R(x₁,... ,x_n) to its standard representation modulo Iq. The basic step in this algorithm is the reduction of terms to polynomials of smaller leading term. In the proof of Proposition 1.10 we can choose k to be the least integer such that j_k > j_k+1, thus yielding a procedure to define for every non-standard monomial λM a reduction denoted red that acts non-trivially on M. In this way, the linear map red : R(xi,..., x_n) → R(xi,..., x_n) depends on M. However, the following procedure is an algorithm.

An element f ∈ R(x₁,... ,x_n) is called normal if deg(X_t) ≤_deglex deg(lt(f)), for every term X_t # lt(f) in f. (In Definition 2.4 we will see that elements of skew PBW extensions are normal).

Proposition 1.12 ([29], Proposition 3.2.4). Let (Q, ≤_deglex) be a skew quantum reduction system. There exists a R-linear map stred_Q: R( x₁ , . . . , x_n ) → R( x₁ , ..., x_n)_irr satisfying the following conditions: (i) for every f ∈ R(x₁ ,...,x_n), there exists a finite sequence r₁,..., r_m of reductions such that stredQ(f) = (r_m ... r₁)(f); (ii) if f is normal, then mdeg(lm(f)) = mdeg(lm(stred_Q(f))).

From the proof of Proposition 1.12 we obtain the next algorithm. Remark 1.13 and Theorem 1.14 are the key results connecting this section with skew PBW extensions.

Remark 1.13. A free left R-module A is a skew PBW extension with respect to ≤_deglex if and only if it is isomorphic to the quotient R(x₁,..., x_n)/I_Q, where Q is a skew reduction system with respect to ≤_deglex.

By Proposition 1.8, the set of all standard terms forms a R-basis for A = R(x₁,..., x_n)/Iq. We have the following key result:

(i) A is a skew PBW extension of R;

(ii) the standard terms form a basis of A as a left free R-module;

(iii) gkji = hkji, for every 1 ≤ i < j < k ≤ n;

(iv) stred_Q(xk fji) = stred_Q(fj xi), for every 1 ≤ i < j < k ≤ n.

Moreover, if A is a skew PBW extension, then stredQ = rQ and A is isomorphic as a left module to R(x1, . . . , xn)irr whose module structure is given by the product f * g := r-Q(fg), for every f,g ∈ R(x₁,...,xⁿ)_irr.

Proof. The equivalence between (i) and (ii) as well between (i) and (iii) is given by Proposition 1.8. The equivalence between (i) and (iv) is obtained from Proposition 1.8 and Proposition 1.12. The remaining statements are also consequences of Proposition 1.8.

Theorem 1.14 gives an algorithm to check whether the algebraic structure R(x₁,... ,x_n)/Iq is a skew PBW extension since stredQ(xkfji) and stredQ(fkjxi) can be computed by means of Algorithm "Reduction to standard form algorithm".

Remark 1.15. In [38], it was also investigated the problem of determining if one quantum algebra have a PBW basis, and more especifically, if the algebra is a skew PBW extension, using different tools. In this sense, our Theorem 1.14 establishes an analogous result to [38], Theorem 2.4.

2. Skew Poincare-Birkhoff-Witt extensions

Definition 2.1 ([18], Definition 1). Let R and A be rings. We say that A is a skew PBW extension of R (also called a a-PBW extension of R), if the following conditions hold:

(i) R ⊆ A;

(ii) there exist elements x₁ , . . . , x_n ∈ A such that A is a left free R-module, with basis the basic elements Mon(A) .

(iii) For each 1 ≤ i ≤ n and any r ∈ R \ {0}, there exists an element c_i,r ∈ R \ {0} such that x_ir - c_i,rx_i ∈ R.

(iv) For any elements 1 ≤ i, j ≤ n, there exists ∈ R \ {0} such that xjx_i - x_ixj ∈ R + Rx₁ + ... + Rx_n.

Under these conditions, we write A := σ(R)(x₁,... ,x_n).

Proposition 2.2 ([18], Proposition 3). Let A be a skew PBW extension of R. For each 1 ≤ i ≤ n, there exists an injective endomorphism σ_i : R → R and an σ_i-derivation δ_i : R → R such that x_ir = σ_i(r)x_i + δ_i(r), for each r ∈ R.

Two particular cases of skew PBW extensions are considered in the following definition.

Definition 2.3 ([18], Definition 4). Let A be a skew PBW extension of R. (a) A is called quasi-commutative if the conditions (iii) and (iv) in Definition 2.1 are replaced by (iii'): for each 1 ≤ i ≤ n and all r ∈ R \ {0} there exists c_i,r ∈ R \ {0} such that xir = ci,rxi; (iv'): for any 1 ≤ i, j ≤ n there exists ci,j ∈ R \ {0} such that x_j x_i = c_i,jx_ix_j; (b) A is called bijective if σ_i is bijective for each 1 ≤ i ≤ n, and c_i,j is invertible for any 1 ≤ i ≤ j ≤ n.

(i)

(ii) For X = x^α ∈ Mon(A), exp(X) := α and deg(X) := |α|. The symbol y will denote a total order defined on Mon(A) (a total order on Nq¹ ). For an element x^α ∈ Mon(A), exp(x^α) := α ∈ Nq¹. If x^a y but x^a = , we write x^a y . Every element f e A can be expressed uniquely as f - Ü0 + 01X1 + ... + a_mX_m, with a_i E R \ {0}, and X_m y ... y X₁. With this notation, we define lm(f) := X_m, the leading monomial of f; lc(f) := a_m, the leading coefficient of f; lt(f) := a_mX_m, the leading term of f; exp(f) := exp(Xm), the order of f; and E(f) := {exp(X_i) | 1 < i < t}. Note that deg(f) := max{deg(X_i)}*=₁. Finally, if f = 0, then lm(0) := 0, lc(0) :=0, lt(0) := 0. We also consider X y 0 for any X E Mon(A). Again, for a detailed description of monomial orders in skew PBW extensions, see [18], Section 3.

3. Examples

In this section we present two examples of skew PBW extensions which illustrate the results of Section 1.1. Our aim is to show that several rings have a PBW basis in the sense of Definition 2.1. Other well known examples for quantum physics (Weyl algebras, quantum Weyl algebras, dispin algebras, Woronowicz algebra, skew polynomial rings, q-Heisenberg algebra, etc) can be realized following the ideas presented in this paper (see [29] for a detailed description of each one of these algebras).

Hayashi algebra

With the purpose of obtaining bosonic representations of the Drinfield-Jimbo quantum algebras, Hayashi considered in [39] the algebra. Let us see its construction (we follow [34], Example 2.7.7). Let us be the algebra generated by the indeterminates , with the relations

Let The relations (3.1) are equivalent to

As we have seen, form a k-basis of U. Now, to obtain the Hayashi algebra , we take the field of the complex numbers and consider the multiplicative monoid S generated by ω_l,..., ω_n. Since S is a regular Ore set and the localization S^-lU exists, then is S^-lU modulo the ideal generated by (see [20], section 3.8, for localizations in skew PBW extensions).

Non-Hermitian realization of a Lie deformed, non-canonical Heisenberg algebra

In [6], it was studied the non-Hermitian realization of a Lie deformed, a non-canonical Heisenberg algebra, considering the case of operators A_j, B_k which are non-Hermitian (i.e., =1)

and,

where . If the operators A_j, B_k are in the form , are leader operators of the usual Heisenberg-Weyl algebra, with N_j the corresponding number operator , and the structure functions f_j(N_j + 1) complex, then it is showed in [6] that A_j and B_k are given by

Then,

Since stred_Q , then the elements , for every i, form a basis of the Lie-deformed Heisenberg algebra, and from (3.2), we can see that this algebra is a skew PBW extension over the complex numbers.

Conclusions and future work

In this paper, we have presented a criteria to determine whether an algebra defined by generators and relations can be expressed as a skew PBW extension. Nevertheless, since the limited size of the paper, there are a lot of remarkable algebras of the theoretical physics which are skew PBW extensions and were not illustrated here (see [29] for more examples). As a future work, we will investigate a theory of PBW bases for another kinds of quantum algebras more general than skew PBW extensions over fields. The techniques to be used will concern noncommutative differential geometry (see [27]) with the aim of characterizing algebras arising in geometries of noncommutative spaces and their interactions with quantum physics, in the sense of [40], [41], and others.

Acknowledgment

The first author is supported by Grant HERMES CODE 30366, Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá.

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