Noncommutative Gröbner bases and filtered-graded transfer.
The book starts with a short history of noncommutative Gröbner bases. There are mentioned papers by B. Buchberger (1965, 1985) as well as A. Galigo (1985), J. Appel and W. Lassner (1988), A. Kandri-Rody and V. Weispfenning (1990), G. Bergman (1978), T. Mora (1986, 1994), E. L. Green (1993). I would like to add to this list the paper by \textit{A. I. Shirshov} [Sib. Mat. Zh. 3, 292-296 (1962; Zbl 0104.26004), translated in ACM SIGSAM Bull. 33, No. 2, 3-9 (1999)]. In this paper Shirshov introduced a concept that is now known as Gröbner-Shirshov bases for Lie polynomials \(\text{Lie}(X)\) over the field \(k\) under Lie bracket \([xy]=xy-yx\). Note that Shirshov was dealing with Lie polynomials via noncommutative (associative) polynomials \(k\langle X\rangle\), \(\text{Lie}(X)\subset k\langle X\rangle\). To be more precise, to make the composition (\(S\)-polynomial by Buchberger's later terminology) of two monic Lie polynomials \(f,g\) relative to some word (ambiguity with Bergman's later terminology) \(w\), where \(w=\overline fv=u\overline g\) (\(\overline f\) the leading associative monomial of \(f\) as noncommutative polynomial), he made first the associative composition \((f,g)_w=fv-ug\), and then he put the special Lie bracketing \([fv]-[ug]\) in order to get the Lie composition. The same was done with the notion of an elimination of leading monomial of one Lie polynomial in other: he made first the elimination in the associative sense and then put the special Lie bracketing. He proved the Composition lemma: Let \(S\) be a set of Lie polynomials, \(S^*\) be a completion of \(S\) under all possible multiple compositions. If \(f\in\text{Id}(S)\) then \(\overline f\) contains as a subword \(\overline s\) for some \(s\in S^*\). Shirshov had further assumed that \(S\) is stable in the sense that for any \(f,g\in S^*\) the degree of a composition \((f,g)_w\) (after eliminations of leading words) is greater then the degrees of \(f,g\). He did not use this condition in the proof. On the other hand stability of \(S\) guarantees that \(S^*\) is a recursive set (for a finite \(S\)). \(S^*\) is known now as Gröbner-Shirshov basis (of \(\text{Id}(S)\)). Of course the Composition lemma for associative polynomials is just a simple version of the case of Lie polynomials (we don't need extra Lie bracketing). In 1972 [\textit{L. A. Bokut'}, Izv. Akad. Nauk SSSR, Ser. Mat. 36, 1173-1219 (1972; Zbl 0252.02046); translation in Math. USSR, Izv. 6 (1972), 1153-1199 (1974)], I formulated Shirshov's Composition lemma in the modern form: Let \(S\) be a set of Lie polynomials that is closed under compositions (any composition goes to zero under elimination of leading words of \(S\), or \(S=S^*\)). If \(f\in\text{Id}(S)\) then \(\overline f\) contains as a subword \(\overline s\) for some \(s\in S\). For commutative polynomials Shirshov's Composition lemma is essentially the same as Buchberger's theorem. For noncommutative polynomials this lemma is essentially the same as Bergman's Diamond lemma. Shirshov's Composition lemma for noncommutative polynomials was explicitly formulated in the paper [\textit{L. A. Bokut'}, Algebra Logika 15, 117-142 (1976; Zbl 0349.16007)]. All in all, Shirshov's (1962) paper must be viewed as the very beginning of Gröbner bases theory not only for Lie polynomials, but also for noncommutative polynomials. Let us turn back to the book. The book contains 8 Chapters. Chapters 1-3 deal with Gröbner bases for (left) ideals in the following setting. Let \(A=k[a_i,\;i\in J]\) be a (noncommutative associative) algebra over the field \(k\) with fixed generators, \(B\) be a fixed set of monomials in \(\{a_i\}\) that form a linear basis of \(A\). Let us fix a (left) monomial order \(u>v \) on \(B\). Then the triple \((A,B,>)\) is called an admissible system. There are many important examples of admissible systems: iterated skew polynomial rings, universal enveloping algebras, different versions of Weyl algebra, quantum groups, solvable polynomial algebras in the sense of Kandri-Rody and Weispfenning (1990), the graded algebra \(G(A)\) and the Rees algebra \(\widetilde A\) of a naturally filtered algebra \(A\), and many others, not speaking about algebras of commutative and noncommutative polynomials. In this context one may naturally define a Gröbner basis of a (left) ideal \(I\) in \(A\). An introduction to Gröbner bases theory of associative algebras is given mostly based on the above mentioned papers by Green, Kandri-Rody and Weispfenning, Mora. The text is written very carefully, so some inaccuracy may appear (such as \(f=\sum c_{i_1,\dots,i_n}a_{i_1}^{\alpha_1}\cdots a_{i_n}^{\alpha_n}\) for EVERY \(f\in A\), see p. 34). The noncommutative version of Buchberger's theorem is proved. (Left) Dickson admissible systems are defined. The title of Chapter 4 (Filtered-graded transfer of Gröbner bases) is included in the title of the book. Here the author studies interconnections between Gröbner bases of \(A\), \(G(A)\), \(\widetilde A\). The typical example is Theorem 4.2.1: Let \((A,B,>_{gr})\) be a (left) admissible system, where \(>_{gr}\) is a graded monomial order (with respect to the natural grading of \(A\)). Suppose that \(G(A)\) is a domain. Let \(I\) be a (left) ideal of \(A\). If \(\{g_i\}\) is a (left) Gröbner basis for \(I\), then \(\{\sigma(g_i)\}\) is a left Gröbner basis for \(G(I)\), where \(\sigma(g)\) is the homogeneous element represented by \(g\) in \(G(A)_n\), \(g\in F_nA-F_{n-1}A\). The converse is also true. An analogous theorem is also valid for the Rees algebra \(\widetilde A\). Graded transfer is also proved for the (left) admissible Dickson systems \(A\) and \(G(A)\). Filtered transfer is known to be valid only in one way (if \((\widetilde A,\widetilde B,>_{gr})\) is a (left) Dickson admissible system, then so is \((A,B,>)\)). The converse is valid for quadric solvable polynomial algebras (Theorem 4.4.2). Chapters 5-8 focus on quadric solvable polynomial algebras. Let us outline the definition. Let \(A=k[{a_1,\dots,a_n}]\) be a finitely generated algebra, \(B=\{a^\alpha=a_1^{\alpha_1}\cdots a_n^{\alpha_n}\}\) be the set of all \textit{standard} monomials that is assumed to be a linear basis of \(A\), \(>_{gr}\) be a graded monomial order on \(B\). Then \(A\) is called a solvable polynomial algebra if \(a^\alpha a^\beta=\lambda_{\alpha,\beta}a^{\alpha+\beta}+f\), where \(\lambda\) is non-zero, and the leading monomial of \(f\) is less than \(a^{\alpha+\beta}\). \(A\) is called a quadric solvable polynomial algebra if in addition the generators of \(A\) satisfy \textit{only} quadric relations. -- Chapter 5 is dealing with the Gel'fand-Kirillov dimension of modules \(M=A/L\), where \(L\) is a left ideal of \(A\), and elimination of variables. It includes the Hilbert polynomial of \(M\), GK-dimension formulas for \(M\) in different cases (\(L\) has a finite Gröbner basis; \(A\) is linear; general case), elimination lemmas for quadric (linear) solvable polynomial algebras. Chapter 6 contains computations of the multiplicity \(e(M)\) of the above module \(M\) defined as \(n!c_d\), where \(c_d\) is the leading coefficient of the Hilbert polynomial of \(M\). There is an application to \(A_n(q_1,\dots,q_n)\), the additive analogue of the Weyl algebra. Chapter 7 deals with the quadric solvable polynomial algebras of type \(A=k[x_1,\dots,x_n,d_1,\dots,d_n]\), where \(x_i,d_i\) satisfy a certain specific system of quadric equations. Then \(A\) can be written as \(A=k[x_1,\dots,x_n][d_1,\dots,d_n]\), where \(k[x_1,\dots,x_n]\) is the polynomial algebra. The author further assumes that \(S=k[x_1,\dots,x_n]-\{0\}\) is a left and right Ore set in \(A\). Then the localization \(S^{-1}A\) has the form \(S^{-1}A=k(x_1,\dots,x_n)[d_1,\dots,d_n]\). A left ideal \(I\) of \(S^{-1}A\) is called \(d\)-finite, if \(S^{-1}A\) is a finite dimensional vector space over \(k(x_1,\dots,x_n)\). The contraction of \(I\) is defined as \(I^c=I\cap A\). It is proved that \(\text{GK.dim}(A/I^c)=n\) for every \(d\)-finite proper left ideal \(I\). Chapter 8 is devoted to the following open problem: Every quadric solvable polynomial algebra \(A\) is Auslander regular. The problem has positive solution for such algebras of \textit{tame} type.
scientific article; zbMATH DE number 1833671