We classify all complex 7- and 8-dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex 9-dimensional dual mock-Lie algebras.
The problem of which Gauss diagram can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows "the number of exits = the number of entrances" and the sufficient condition is based on Jordan curve Theorem. Further, using matrixes we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.Mathematics Subject Classifications: 57M25, 14H50.
In this paper we calculate the cohomology ring Ext * kPln (k, k) and the Hochschild cohomology ring of the plactic monoid algebra kPln via the Anick resolution using a Gröbner -Shirshov basis. IntroductionThe plactic monoid was discovered by Knuth [15], who used an operation given by Schensted in his study of the longest increasing subsequence of a permutation. It was named and systematically studies by Lascoux and Schützenberger [22], who allowed any totally ordered alphabet in the definition. It is known that the elements of plactic monoid can be written in the canonical form, and in this form can be identified with some type of the Young tableaux. Because of its strong relations to Young tableaux, the plactic monoid has already become a classical tool in several areas of representation theory and algebraic combinatorics [20].Among the significant applications are: a proof of the Littlewood-Richardson rule, an algorithm which allows to decompose tensor product of representations of unitary groups, a combinatorial description of the Kostka -Foulkes polynomials, which arise as entries of character table of the finite linear group. The plactic monoid appeared also in: theory of modular representations of the symmetric group, quantum groups, via the representation theory of quantum enveloping algebras. It is worth mentioning that even though the combinatorics of the plactic monoid has been extensively studied, there are only a few preliminary results of the corresponding plactic algebra over a field [9].The plactic monoid was connected to the free 2-nilpotent Lie algebra (which is a subalgebra of the parastatistics algebra) in the work [24] by J-L. Loday and T. Popov. The connection is through the quantum deformation (in the sense of Drinfeld) of the parastatistics algebra. But the first work where the connection between the plactic monoid algebra and parastastics algebra (in two dimensions) was found is [30].In [5] there was given an independent proof of uniqueness of (Robinson -Shensted) Knuth normal forms of elements of (Knuth -Schützenberger) plactic monoid.Gröbner bases and Gröbner -Shirshov bases were invented independently by A.I. Shirshov for ideals of free (commutative, anti-commutative) non-associative algebras [17,28], free Lie algebras [27,28] by H. Hironaka [13] for ideals of the power series algebras (both formal and convergent), and by B. Buchberger [6] for ideals of the polynomial algebras.Cain et al [7] use the Schensted-Knuth normal form (the set of (semistandart) Young tableaux) to prove that the multiplication table of column words (strictly decreasing words with respect to some order on the letters) forms a finite Gröbner -Shirshov basis of the finitely generated plactic monoid. In [5] were given new explicit formulas for the multiplication tables of row (nondecreasing word) and column words and independent proofs that the resulting sets of relations are Gröbner -Shirshov bases in row and column generators respectively.The Anick resolution was obtained by David J. Anick in 1986 [1]. This is a r...
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