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We prove that if, for a non‐trivial variety of non‐associative algebras, every subalgebra of every free algebra is free and I2$I^2$ is an ideal whenever I$I$ is an ideal, then this variety coincides with the variety of all Lie algebras. [ABSTRACT FROM AUTHOR]

We develop the deformation theory of cohomological field theories (in short, CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduce two new natural extensions of the notion of a CohFT: homotopy (necessary algebraic toolkit to develop chain-level Gromov–Witten invariants) and quantum (with examples found in the works of Buryak and Rossi on integrable systems). The universal group of symmetries of morphisms of modular operads, based on Kontsevich's graph complex, is shown to be trivial. Using the tautological rings on moduli spaces of curves, we introduce a natural enrichment of Kontsevich's graph complex. This leads to universal groups of non-trivial symmetries of both homotopy and quantum CohFTs, which, in the latter case, is shown to contain both the prounipotent Grothendieck–Teichmüller group and the Givental group. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the Poincaré-Birkhoff-Witt (PBW) theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of Kähler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra. [ABSTRACT FROM AUTHOR]

To the memory of V. A. Artamonov (1946–2021) All algebras of a certain type are said to form a Nielsen–Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen–Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen–Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well-known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen–Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen–Schreier varieties of algebras with one binary operation and identities of degree four. [ABSTRACT FROM AUTHOR]

The Mn4+-doped α-Al2O3 with particle size of 70-600 nm was obtained by combustion method. Based on the results of luminescence measurements, the crystal-field strength (Dq) and Racah parameters (B, C) for Mn4+ in α-Al2O3 were determined using a pure electronic transition approach. The obtained values of Dq (1898 cm-1) and nephelauxetic parameter β1 (0.95) for Mn4+ ions in α-Al2O3 are consistent with those of Mn4+ in other oxide compounds. In particular, the Dq/B value of 2.77 indicates a strong crystal field environment of the Mn4+ ions in the α-Al2O3 lattice. A comparison with literature data for Cr3+ in α-Al2O3 was also carried out. The composites of general formula α-Al2O3: Mn4+, Mg2+/Gd3Al5O12:Ce3+ had been also obtained. It is shown that these materials demonstrate the intense broadband emission with maxima at about 585 and 678 nm. The broad band with a maximum at 585 nm is caused by the 5d→4f transitions of Ce3+ ions in Gd3Al5O12, whereas the band at 678 nm is caused by the Mn4+ 2Eg→4A2g transitions in α-Al2O3. The emission color changes from yellow to deep red with increasing content of α-Al2O3:Mn4+, Mg2+, and the luminescence quantum efficiency of the composites was found as high as 0.60. [ABSTRACT FROM AUTHOR]

The experimental results for chip-to-chip data communications on a superconducting multi-chip-module (MCM) using a novel fabrication technique, which successfully passed single-flux quanta at rates exceeding 50 Gbps, is reported.

In this paper, we study the C-enriched pre-Lie operad defined by Calaque and Willwacher for any Hopf cooperad C to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding enriched pre-Lie operads; we prove criteria for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré-Birkhoff-Witt type theorem for universal enveloping brace algebras of pre-Lie algebras. [ABSTRACT FROM AUTHOR]

Using methods of computer algebra, especially, Gröbner bases for submodules of free modules over polynomial rings, we solve a classification problem in theory of algebraic operads: we show that the only nontrivial (possibly inhomogeneous) distributive law between the operad of Lie algebras and the operad of commutative associative algebras is given by the Livernet–Loday formula deforming the Poisson operad into the associative operad. [ABSTRACT FROM AUTHOR]

A series of tris(pyrazolyl)borate mono-, di- and trinuclear complexes, [Tp2Ln]nX (Ln = Eu, Tb, Gd, Dy, Xn− = various mono-, bis- and tris(β-diketonates) has been prepared. The Tb3+ and Dy3+ complexes are luminescent single molecular magnets (SMM) and exhibit luminescence quantum efficiencies up to 73% for the Tb3+ and 4.4% for the Dy3+ compounds. Similar Eu3+ complexes display bright emission only at lower temperatures. The Dy3+ and Tb3+ complexes possess SMM behavior in a non-zero dc field at low temperatures, while the polynuclear Dy3+ complexes also show slow magnetic relaxation even in zero dc field up to 8 K. Ueff-values determined from dynamic magnetic measurements were up to 31 and 6 cm−1 for the Dy3+ and Tb3+ complexes, respectively. It was found that within a series of Dy3+ and Tb3+ compounds, Ueff and luminescence quantum yields decreased with increasing nuclearity of the compounds and a shortening of the intramolecular Ln–Ln distance. ΔOrbach-values estimated from low-temperature luminescence spectra were significantly higher than those obtained from ac magnetic data, which may be due to involvement of additional processes in the relaxation mechanism (quantum tunneling, Raman, direct) reducing the energy barrier. Some of the Tb3+-compounds also display metal-centred electroluminescence, giving them potential as emitting layers in LEDs. [ABSTRACT FROM AUTHOR]

Using the categorical approach to Poincaré–Birkhoff–Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota–Baxter algebras of tridendriform algebras, for universal enveloping Rota–Baxter Lie algebras of post-Lie algebras, and for universal enveloping tridendriform algebras of post-Lie algebras. Similar results, though without functoriality of the PBW isomorphisms, were recently obtained by Gubarev. Our methods are completely different and mainly rely on methods of rewriting theory for shuffle operads. Communicated by Pavel Kolesnikov [ABSTRACT FROM AUTHOR]

To the memory of Trevor Evans (1925–1991), the pioneer of interchange laws in universal algebra We establish a combinatorial model for the Boardman–Vogt tensor product of several absolutely free operads, that is, free symmetric operads that are also free as -modules. Our results imply that such a tensor product is always a free -module, in contrast with the results of Kock and Bremner–Madariaga on hidden commutativity for the Boardman–Vogt tensor square of the operad of non-unital associative algebras. [ABSTRACT FROM AUTHOR]

We construct for a forgetful functor from nonsymmetric operads to weight graded associative algebras, two functorial "enveloping operads," the right inverse and the left adjoint of the forgetful functor. Those functors turn out to be related by operadic Koszul duality, and that relationship can be utilized to exhibit an example of a binary quadratic non-Koszul operad for which the Ginzburg–Kapranov functional equation holds and an example of a binary quadratic Koszul operad for which the Koszul property cannot be established using Gröbner bases. Both examples resolve standing open problems in operad theory. We also apply the enveloping operad functors to get a new proof of the Lagrange inversion formula using homotopical algebra. [ABSTRACT FROM AUTHOR]

We introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2‐discs, framed little 2‐discs, and Deligne–Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental‐type symmetries. This relies on rich geometry of the analogues of the Deligne–Mumford spaces, coming from the fact that they admit several equivalent interpretations: as the toric varieties of Loday's realisations of the associahedra, as the brick manifolds recently defined by Escobar, and as the De Concini–Procesi wonderful models for certain subspace arrangements. [ABSTRACT FROM AUTHOR]

We show how to study a certain associative algebra recently discovered by Iyudu and Shkarin using the Anick resolution. This algebra is a counterexample to the conjecture of Positselski on Koszul algebras of finite global dimension. [ABSTRACT FROM PUBLISHER]