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Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

The "similarity degree" of a unital operator algebra A was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type II1 factor with property Γ is at most 5, improving on a previous bound (≤ 44) due to E. Christensen.

Doty and Giaquinto constructed a certain set denoted by [Formula: see text] in this paper, and conjectured that the set [Formula: see text] forms a [Formula: see text]-basis for the integral [Formula: see text]-Schur algebra [Formula: see text], where [Formula: see text]. In this paper, we show that the set [Formula: see text] is a [Formula: see text]-basis of [Formula: see text]-Schur algebra [Formula: see text] over [Formula: see text]. The set [Formula: see text] is a truncated form of PBW basis for quantum [Formula: see text]. Moreover, we give an example to show that the set [Formula: see text] is not [Formula: see text]-basis of [Formula: see text] in general.

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

This paper derives several new results, as well as gives a partial survey of the recent development, on the value sharing for algebraic and holomorphic curves into [Formula: see text].

In this paper, we construct explicitly all dually flat Randers metrics by using the bijection between Randers metrics and their navigation representation.

In this paper, we generalize Izumi’s result on uniqueness of realization of finite C[Formula: see text]-tensor categories in the endomorphism category of the injective factor of type III1 for finitely generated strongly amenable C[Formula: see text]-tensor categories by applying Popa’s classification theorem of strongly amenable subfactors of type III1.

In this paper, we study a generalization of twisted (groupoid) equivariant K-theory in the sense of Freed–Moore for [Formula: see text]-graded [Formula: see text]-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele’s K-theory for [Formula: see text]-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant K-group when the [Formula: see text]-algebra is trivially graded. It is applied for the bulk-edge correspondence of topological insulators with CT-type symmetries.

In this paper, we study the structure theory of two classes of Lie superalgebras [Formula: see text] of Block type, where [Formula: see text] is a positive integer. In particular, the automorphism groups, derivation superalgebras and central extensions of [Formula: see text] are completely determined.