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Extra resources for Algebra VI: Combinatorial and Asymptotic Methods of Algebra. Nonassociative Structures (Encyclopaedia of Mathematical Sciences)
Theorem 4. Let d = (2; 1, l/r;n, n, . . n). 7182.. , then the condition (A) holds. If T < 2”/e(n - l), then the condition (F) holds. For T = 4, n = 6, the implication (0) =+ (A) does not hold. A proof is in the same papers. A counterexample to the implication is the algebra (z, y 1 ~~yzy~,~~y~zy,z~y~, z2y4 - zyzyzy). According to the composition lemma, only the element z2yzyzy is missing for a GrZjbner basis. 7: (1 - 2t + (4t6 + t7) - t’)-1 = Pd(t). Theorem 5. Let d = (g; 1, 1, . . , l/r; 2,.
E. the algebras A E Gd are quadratic - cf. 7). Conditions (A)-(F) of Th eorem 3 are mutually equivalent and are equivalent to the condition 4T < g2. Proof. e. 4~ < g2. On the other hand, the last condition implies the condition (B), according to the example considered after Theorem 2 (in the case of even g). For the case of odd g we have Thus, if there is an algebra in Gd, for which HA = [Pdl, then that, series will be standard. If there are sufficiently many relations, this is exactly the case.
X,, ~1,. . , ym and choose R to be the set of of the elements of the form xiyj. This set will obviously be combinatorially free and consequently the algebra will be standard and the standard Hilbert series will be equal to (1 - 2mt + rt2)-l. All the standard algebras for this vector have global dimension two. Theorem 3. Let us consider the following six statements, formulated for a given degrees vector d: (A) There exist two polynomials F(t) and G(t) with non-negative integer coeficients, such that (1 - F)(l - G) 2 1 - Hx + HR = Pi?