We prove some commutation relations for a 3-graded Lie algebra, i.e., a Z-graded Lie algebra whose nonzero homogeneous elements have degrees −1, 0 or 1, over a field K. In particular, we examine the free 3-graded Lie algebra generated by an element of degree −1 and another of degree 1. We show that if K has characteristic zero, such a Lie algebra can be realized as a Lie algebra of matrices ove...

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $$L_K(\mathcal {G})$$ of an ultragraph $$\mathcal {G}$$ over a field K is purely infinite simple that it von Neumann regular. Consequently, obtain every graded either locally matricial algebra, or full matrix ring $$K[x, x^{-1}]$$ , algebra.

Green and Marcos [3] call a graded k-algebra δ-Koszul if the corresponding Yoneda algebra Ext(k, k) is finitely generated and Ext(k, k) is zero unless j = δ(i) for some function δ : N→ N. For any integer m ≥ 3 we exhibit a non-commutative quadratic δ-Koszul algebra for which the Yoneda algebra is generated in degrees (1, 1) and (m,m+ 1). These examples answer a question of Green and Marcos. The...

We show that the Nakayama automorphism of a Frobenius algebra R over a field k is independent of the field (Theorem 4). Consequently, the k-dual functor on left R-modules and the bimodule isomorphism type of the k-dual of R, and hence the question of whether R is a symmetric k-algebra, are independent of k. We give a purely ring-theoretic condition that is necessary and sufficient for a finite-...

Let P ⊃ ∞. Recent developments in commutative arithmetic [14, 14] have raised the question of whether there exists a hyper-Euclidean, Déscartes and Möbius compactly dependent, regular, Weierstrass algebra. We show that S is less than p. In this context, the results of [14] are highly relevant. A central problem in arithmetic potential theory is the construction of extrinsic, Σ-algebraic functio...

‎Let $lambda_1,dots,lambda_n$  be positive real numbers such that $sum_{k=1}^n lambda_k=1$. In this paper, we prove that for any positive operators $a_1,a_2,ldots, a_n$ in semifinite von Neumann algebra $M$ with faithful normal trace that $t(1)

We describe an $E_k$-cell structure on the free $E_{k+1}$-algebra a point, and more generally how May-Milgram filtration of $\Omega^m \Sigma^m S^{k}$ lifts to $E_{k+m}$-algebra point by iterated pushouts $E_k$-algebras.

We show that the functor of $p$-typical co-Witt vectors on commutative algebras over a perfect field $k$ characteristic $p$ is defined on, and in fact only depends weaker structure than $k$-algebra. call this $p$-polar By extension, functors points for any $p$-adic affine group scheme formal are depend structures. In terms abelian Hopf algebras, we cofree cocommutative algebra can be $k$-algebr...

We study automorphisms φ of the free associative algebra K〈x, y, z〉 over a field K such that φ(x), φ(y) are linear with respect to x, y and φ(z) = z. We prove that some of these automorphisms are wild in the class of all automorphisms fixing z, including the well known automorphism discovered by Anick, and show how to recognize the wild ones. This class of automorphisms induces tame automorphis...

Automated reasoning systems often su er from redundancy: similar parts of derivations are repeated again and again. This leads us to the problem of loopdetection, which clearly is undecidable in general. Nevertheless, we tackle this problem by extending the hyper-tableau calculus as proposed in [Baumgartner, 1998] by generalized terms with exponents, that can be computed by means of computer al...