Graded A-identities for M 1,1 (E) Augusto Naves, Fernando; Luiz Talpo, Humberto
Linear & multilinear algebra,
12/19/2022, Letnik:
70, Številka:
20
Journal Article
Recenzirano
Let F be a field of characteristic 0, E be the unitary infinite dimensional Grassmann algebra over F and consider the algebra
with its natural
-grading. We describe the graded A-identities for
and we ...compute its graded A-codimensions.
We review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In ...particular, we show how a field theory gives rise to an L∞‐algebra and how quasi‐isomorphisms between L∞‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern–Simons theories and give some useful shortcuts in usually rather involved computations.
The authors review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, it is shown how a field theory gives rise to an L∞‐algebra and how quasi‐isomorphisms between L∞‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. Most importantly, higher (categorified) Chern–Simons theories are discussed in great detail and some useful shortcuts in usually rather involved computations are given.
Gradings on block-triangular matrix algebras Diniz, Diogo; Galdino da Silva, José; Koshlukov, Plamen
Proceedings of the American Mathematical Society,
01/2024, Letnik:
152, Številka:
1
Journal Article
Recenzirano
Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial ...identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura Arch. Math. (Basel) 110 (2018), pp. 327–332 this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev Arch. Math. (Basel) 89 (2007), pp. 33–40.
The defect formula Nart, Enric; Novacoski, Josnei
Advances in mathematics (New York. 1965),
09/2023, Letnik:
428
Journal Article
Recenzirano
In this paper we present a characterization for the defect of simple algebraic extensions of valued fields. This characterization generalizes the known result for the henselian case, namely that the ...defect is the product of the relative degrees of limit augmentations. The main tool used here is the graded algebra associated to a valuation on a polynomial ring. Let Kh be a henselization of a valued field K. Another relevant result proved in this paper is that for every valuation μh on Khx, with restriction μ on Kx, the corresponding map Gμ↪Gμh of graded algebras is an isomorphism.
We study strongly graded groupoids, which are topological groupoids G equipped with a continuous, surjective functor κ:G→Γ, to a discrete group Γ, such that κ−1(γ)κ−1(δ)=κ−1(γδ), for all γ,δ∈Γ. We ...introduce the category of graded G-sheaves, and prove an analogue of Dade's Theorem: G is strongly graded if and only if every graded G-sheaf is induced by a Gε-sheaf. The Steinberg algebra of a graded ample groupoid is graded, and we prove that the algebra is strongly graded if and only if the groupoid is. Applying this result, we obtain a complete graphical characterisation of strongly graded Leavitt path and Kumjian-Pask algebras.
In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some ...multilinear polynomial is a central polynomial, or a trace zero polynomial, and we use this approach to present an equivalent statement to the Lvov-Kaplansky conjecture.
On sums of gr-PI algebras Fagundes, Pedro; Koshlukov, Plamen
Linear algebra and its applications,
11/2023, Letnik:
677
Journal Article
Recenzirano
Let A=B+C be an associative algebra graded by a group G, which is a sum of two homogeneous subalgebras B and C. We prove that if B is an ideal of A, and both B and C satisfy graded polynomials ...identities, then the same happens for the algebra A. We also introduce the notion of graded semi-identity for the algebra A graded by a finite group and we give sufficient conditions on such semi-identities in order to obtain the existence of graded identities on A. We also provide an example where both subalgebras B and C satisfy graded identities while A=B+C does not. Thus the theorem proved by Kȩpczyk in 2016 does not transfer to the case of group graded associative algebras. A variation of our example shows that a similar statement holds in the case of group graded Lie algebras. We note that there is no known analogue of Kȩpczyk's theorem for Lie algebras.
Resco and Small gave the first example of an affine Noetherian algebra which is not finitely presented. It is shown that their algebra has no finite-dimensional filtrations whose associated graded ...algebras are Noetherian, affirming their prediction. A modification of their example yields countable fields over which ‘almost all’ (that is, a co-countable continuum of) affine Noetherian algebras lack such a filtration, and an answer to a question suggested by Irving and Small is derived.
In the present paper we study UT(D1,…,Dn), a G-graded algebra of block triangular matrices where G is a group and the diagonal blocks D1,…,Dn are graded division algebras. We prove that any two such ...algebras are G-isomorphic if and only if they satisfy the same graded polynomial identities. We also discuss the number of different isomorphism classes obtained by varying the grading and we exhibit its connection with the factorability of the T-ideal of graded identities. Moreover we give some results about the generators of the graded polynomial identities for these algebras. In particular we generalize the results about the graded identities of UTn to the case in which the diagonal blocks D1,…,Dn are all isomorphic.
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