Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHilbert series of PI relatively free G-graded algebras are rational functions
383
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}s variables that correspond to the g_is, i=1,...,r. Let F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F<x_{1,g_1},...,x_{r,g_r}> which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F<x_{1,g_1},...,x_{r,g_r}>/I is a rational function.
rate research
Read More
It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.
Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. Consider $G$-graded simple algebras $A$ which are finite dimensional and $e$-central over $F$, i.e. $Z(A)_{e} := Z(A)cap A_{e} = F$. For any such algebra we construct a textit{generic} $G$-graded algebra $mathcal{U}$ which is textit{Azumaya} in the following sense. $(1)$ textit{$($Correspondence of ideals$)$}: There is one to one correspondence between the $G$-graded ideals of $mathcal{U}$ and the ideals of the ring $R$, the $e$-center of $mathcal{U}$. $(2)$ textit{Artin-Procesi condition}: $mathcal{U}$ satisfies the $G$-graded identities of $A$ and no nonzero $G$-graded homomorphic image of $mathcal{U}$ satisfies properly more identities. $(3)$ textit{Generic}: If $B$ is a $G$-graded algebra over a field then it is a specialization of $mathcal{U}$ along an ideal $mathfrak{a} in spec(Z(mathcal{U})_{e})$ if and only if it is a $G$-graded form of $A$ over its $e$-center. We apply this to characterize finite dimensional $G$-graded simple algebras over $F$ that admit a $G$-graded division algebra form over their $e$-center.
Let g be a free brace algebra. This structure implies that g is also a prelie algebra and a Lie algebra. It is already known that g is a free Lie algebra. We prove here that g is also a free prelie algebra, using a description of g with the help of planar rooted trees, a permutative product, and anipulations on the Poincare-Hilbert series of g.
Let G be any group and F an algebraically closed field of characteristic zero. We show that any G-graded finite dimensional associative G-simple algebra over F is determined up to a G-graded isomorphism by its G-graded polynomial identities. This result was proved by Koshlukov and Zaicev in case G is abelian.
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
