In this article, we realize the finite range ultragraph Leavitt path algebras as Steinberg algebras. This realization allows us to use the groupoid approach to obtain structural results about these ...algebras. Using the skew product of groupoids, we show that ultragraph Leavitt path algebras are graded von Neumann regular rings. We characterize strongly graded ultragraph Leavitt path algebras and show that every ultragraph Leavitt path algebra is semiprimitive. Moreover, we characterize irreducible representations of ultragraph Leavitt path algebras. We also show that ultragraph Leavitt path algebras can be realized as Cuntz-Pimsner rings.
It has been recently proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed ⁎-graded exponent if, ...and only if, it is generated by a subalgebra of an upper block triangular matrix algebra, A:=UTZ2⁎(A1,…,Am), equipped with a suitable elementary Z2-grading and graded involution. Here we give necessary and sufficient conditions so that IdZ2⁎(A) factorizes in the product of the ideals of ⁎-graded polynomial identities of its ⁎-graded simple components Ai.
There is a longstanding conjecture by Fröberg about the Hilbert series of the ring R∕I, where R is a polynomial ring, and I an ideal generated by generic forms. We prove this conjecture true in the ...case when I is generated by a large number of forms, all of the same degree. We also conjecture that an ideal generated by m'th powers of generic forms of degree d≥2 gives the same Hilbert series as an ideal generated by generic forms of degree md. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.
We introduce Morita and Rickard equivalences over a group graded G-algebra between block extensions. A consequence of such equivalences is that Späth's central order relation holds between two ...corresponding character triples.
The research is motivated by a construction of groups of oscillating growth by Kassabov and Pak 25 and a description of possible growth functions of finitely generated associative algebras by Bell ...and Zelmanov 9. In this paper we address both, the question of possible growth functions in case of Lie algebras, and the Kurosh problem, because our examples of restricted Lie algebras have a nil p-mapping, which is an analogue of nillity for associative algebras or periodicity for groups.
Namely, for any field of positive characteristic, we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is “almost dying” by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is one, more precisely, the growth is of type n(ln⋯ln︸qtimesn)κ, where q∈N, κ>0 are constants. On the other hand, for infinitely many n the growth function has a rather fast intermediate behavior of type exp(n/(lnn)λ), λ being a constant determined by characteristic, for such periods the algebra is “resuscitating”. Moreover, the growth function is bounded and oscillating between these two types of behavior. These restricted Lie algebras have a nil p-mapping, thus addressing the Kurosh problem as well.
Let F be an algebraically closed field of characteristic zero and let G be a finite group. We consider graded Verbally prime T-ideals in the free G-graded algebra. It turns out that equivalent ...definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime G-graded T-ideals and strongly verbally prime T-ideals. At first, following Kemer's ideas, we classify G-graded verbally prime T-ideals. The main bulk of the paper is devoted to the stronger notion. We classify G-graded strongly verbally prime T-ideals which are T-ideal of affine G-graded algebras or equivalently G-graded T-ideals that contain a Capelli polynomial. It turns out that these are precisely the T-ideal of G-graded identities of finite dimensional G-graded, central over F (i.e. Z(A)e=F) which admit a G-graded division algebra twisted form over a field k which contains F or equivalently over a field k which contains enough roots of unity (e.g. a primitive n-root of unity where n=ord(G)).
We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group.
In this paper, we study the images of multilinear graded polynomials on the graded algebra of upper triangular matrices
$UT_n$
. For positive integers
$q\leq n$
, we classify these images on
$UT_{n}$
...endowed with a particular elementary
${\mathbb {Z}}_{q}$
-grading. As a consequence, we obtain the images of multilinear graded polynomials on
$UT_{n}$
with the natural
${\mathbb {Z}}_{n}$
-grading. We apply this classification in order to give a new condition for a multilinear polynomial in terms of graded identities so that to obtain the traceless matrices in its image on the full matrix algebra. We also describe the images of multilinear polynomials on the graded algebras
$UT_{2}$
and
$UT_{3}$
, for arbitrary gradings. We finish the paper by proving a similar result for the graded Jordan algebra
$UJ_{2}$
, and also for
$UJ_{3}$
endowed with the natural elementary
${\mathbb {Z}}_{3}$
-grading.
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