43 research outputs found
The three graces in the Tits--Kantor--Koecher category
A metaphor of Jean-Louis Loday describes Lie, associative, and commutative
associative algebras as ``the three graces'' of the operad theory. In this
article, we study the three graces in the category of -modules
that are sums of copies of the trivial and the adjoint representation. That
category is not symmetric monoidal, and so one cannot apply the wealth of
results available for algebras over operads. Motivated by a recent conjecture
of the second author and Mathieu, we embark on the exploration of the extent to
which that category ``pretends'' to be symmetric monoidal. To that end, we
examine various homological properties of free associative algebras and free
associative commutative algebras, and study the Lie subalgebra generated by the
generators of the free associative algebra.Comment: 17 pages, comments are welcom
Gröbner–Shirshov basis of the Adyan extension of the Novikov group
AbstractThe goal of this paper is to give a comparatively short and simple analysis of the Adyan origional group constraction (S.I. Adyan, Unsolvability of some algorithmic problems in the theory of groups, Trudy MMO 6 (1957) 231–298)
Finite convergent presentations of plactic monoids for semisimple lie algebras
We study rewriting properties of the column presentation of plactic monoid
for any semisimple Lie algebra such as termination and confluence. Littelmann
described this presentation using L-S paths generators. Thanks to the shapes of
tableaux, we show that this presentation is finite and convergent. We obtain as
a corollary that plactic monoids for any semisimple Lie algebra satisfy
homological finiteness properties
Koszul algebras and Donaldson-Thomas invariants
For a given symmetric quiver , we define a supercommutative quadratic
algebra whose Poincar\'e series is related to the motivic
generating function of by a simple change of variables. The Koszul duality
between supercommutative algebras and Lie superalgebras assigns to the algebra
its Koszul dual Lie superalgebra . We prove
that the motivic Donaldson-Thomas invariants of the quiver may be computed
using the Poincar\'e series of a certain Lie subalgebra of
that can be described, using an action of the first Weyl algebra on
, as the kernel of the operator . This gives a new
proof of positivity for motivic Donaldson--Thomas invariants. In addition, we
prove that the algebra is numerically Koszul for every
symmetric quiver and conjecture that it is in fact Koszul; we also prove
this conjecture for quivers of a certain class.Comment: 25 pages, the main result on DT invariants of symmetric quivers is
now not conditional on Koszulnes
Finite Gr\"obner--Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids
This paper shows that every Plactic algebra of finite rank admits a finite
Gr\"obner--Shirshov basis. The result is proved by using the combinatorial
properties of Young tableaux to construct a finite complete rewriting system
for the corresponding Plactic monoid, which also yields the corollaries that
Plactic monoids of finite rank have finite derivation type and satisfy the
homological finiteness properties left and right . Also, answering a
question of Zelmanov, we apply this rewriting system and other techniques to
show that Plactic monoids of finite rank are biautomatic.Comment: 16 pages; 3 figures. Minor revision: typos fixed; figures redrawn;
references update
Properties of the Toric Rings of a Chordal Bipartite Family of Graphs
This project concerns the classification and study of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain ladder-like structures). We determine a linear system of parameters for each ring and explicitly determine the Hilbert series for the resulting Artinian reduction. As corollaries, we obtain the multiplicity and regularity of the original rings. This work extends results known for a subfamily coming from a two-sided ladder and includes constructive proofs which may be useful in future study of these rings and others. We also develop explicit elements in the Priddy complex which correspond via known isomorphisms to Tate variables in the acyclic closure of the residue field over the localization of our rings at their homogeneous maximal ideals
Hochschild Cohomology Of Short Gorenstein Rings
Let k be a field of characteristic 0. In this thesis, we show that the Hochschild cohomology of the family of short Gorenstein k-algebras
sGor(N) =k[X_0,...,X_N](X_iX_j, X_i^2−X_j^2 | i,j= 0,...,N, i different from j), N≥2,
exhibits exponential growth. The proof uses Gröbner-Shirshov basis theory and along the way we describe an explicit monomial basis for the Koszul dual of sGor(N) for N≥2
Some generalizations of the variety of transposed Poisson algebras
It is shown that the variety of transposed Poisson algebras coincides with
the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is
commutative. The Grobner-Shirshov basis for the transposed Poisson operad is
calculated up to degree 4. Furthermore, we demonstrate that transposed Poisson
algebras are F-manifolds. We verify that the special identities of the variety
of GD-algebras hold in transposed Poisson algebras. Finally, we propose a
conjecture stating that transposed Poisson algebras are special, i.e., can be
embedded into a differential Poisson algebra.Comment: 7 p, changed the titl