Recent zbMATH articles in MSC 17https://zbmath.org/atom/cc/172024-02-28T19:32:02.718555ZWerkzeugThe partial Temperley-Lieb algebra and its representationshttps://zbmath.org/1527.051742024-02-28T19:32:02.718555Z"Doty, Stephen"https://zbmath.org/authors/?q=ai:doty.stephen-r"Giaquinto, Anthony"https://zbmath.org/authors/?q=ai:giaquinto.anthonySummary: In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley-Lieb algebra, arising as the generic centralizer algebra \(\mathrm{End}_{\mathcal{U}_q (\mathfrak{gl}_2)} (V^{\otimes k})\), where \(V=V(0) \oplus V(1)\) is the direct sum of the trivial and natural module for the quantized enveloping algebra \(\mathcal{U}_q (\mathfrak{gl}_2)\). It is a proper subalgebra of the Motzkin algebra (the \(\mathcal{U}_q (\mathfrak{sl}_2)\)-centralizer) of \textit{G. Benkart} and \textit{T. Halverson} [Eur. J. Comb. 36, 473--502 (2014; Zbl 1284.05333)]. We prove a version of Schur-Weyl duality for the new algebras, and describe their generic representation theory.New structure on the quantum alcove model with applications to representation theory and Schubert calculushttps://zbmath.org/1527.051752024-02-28T19:32:02.718555Z"Kouno, Takafumi"https://zbmath.org/authors/?q=ai:kouno.takafumi"Lenart, Cristian"https://zbmath.org/authors/?q=ai:lenart.cristian"Naito, Satoshi"https://zbmath.org/authors/?q=ai:naito.satoshiSummary: The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart-Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang-Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang-Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant \(K\)-group of semi-infinite flag manifolds. The generalized quantum Yang-Baxter moves give rise to a ``sijection'' (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula.Chiral de Rham complex on the upper half plane and modular formshttps://zbmath.org/1527.110372024-02-28T19:32:02.718555Z"Dai, Xuanzhong"https://zbmath.org/authors/?q=ai:dai.xuanzhongA central object of study in this paper is the chiral de Rham complex on the upper half-plane \(\mathbb{H}\), and in particular, the global section \(\Omega^{ch}(\mathbb{H})\). As is explained in the paper, there is an \(\operatorname{SL}(2,\mathbb{R})\)-action on \(\Omega^{ch}(\mathbb{H})\) induced by the fractional linear transformations on \(\mathbb{H}\). Taking certain subgroups \(\Gamma\) of \(\operatorname{SL}(2,\mathbb{R})\), one can consider the main object of study in this paper, which is the subspace \(\Omega^{ch}(\mathbb{H},\Gamma)\) of \(\Omega^{ch}(\mathbb{H})\) given by the elements that are \(\Gamma\)-invariant and also holomorphic at the cusps. In particular, the paper takes \(\Gamma\) to be \(\operatorname{SL}(2,\mathbb{Z})\) as well as congruence subgroups. In fact, \(\Omega^{ch}(\mathbb{H})\) has the structure of a vertex operator algebra, and \(\Omega^{ch}(\mathbb{H},\Gamma)\) is a vertex operator subalgebra.
The main results of the paper are then to (i) construct a basis of \(\Omega^{ch}(\mathbb{H},\Gamma)\), (ii) use this basis to obtain the character formula for \(\Omega^{ch}(\mathbb{H},\Gamma)\), and (iii) explain how the vertex operator product \(x_{(n)}y\) for \(x,y\in \Omega^{ch}(\mathbb{H},\Gamma)\) can be expressed in terms of modified Rankin-Choen brackets of modular forms. In addition, the paper shows \(\Omega^{ch}(\mathbb{H},\Gamma)\) is a topological vertex algebra, obtains a chain of vertex algebra ideals in \(\Omega^{ch}(\mathbb{H},\Gamma)\) from which a simple topological vertex algebra can be defined, and also develops results concerning Hecke operators on elements in \(\Omega^{ch}(\mathbb{H},\Gamma)\). It is worth noting that the results obtained in this paper establishing a connection between vertex operator products and Rankin-Cohen brackets verifies a relationship that has long been hypothesized.
Reviewer: Matthew Krauel (Sacramento)On 2nd-stage quantization of quantum cluster algebrashttps://zbmath.org/1527.130222024-02-28T19:32:02.718555Z"Li, Fang"https://zbmath.org/authors/?q=ai:li.fang"Pan, Jie"https://zbmath.org/authors/?q=ai:pan.jieSummary: Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the 2nd-stage quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its 2nd-stage quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses a mutually alternating quantum cluster algebra such that their 2nd-stage quantization can be essentially the same.
As an example, we give the 2nd-stage quantized cluster algebra \(A_{p,q}(SL(2))\) of \(Fun_{\mathbb{C}}(SL_q(2))\) in {\S}7.1 and show that it is a non-trivial 2nd-stage quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. As another example, we present a class of quantum cluster algebras with coefficients which possess a non-trivial 2nd-stage quantization. In particular we obtain a class of quantum cluster algebras from surfaces with coefficients which possess non-trivial 2nd-stage quantization. Finally, we prove that the compatible Poisson structure of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the 2nd-stage quantization of a quantum cluster algebra without coefficients is in fact trivial.Formal Bott-Thurston cocycle and part of a formal Riemann-Roch theoremhttps://zbmath.org/1527.140152024-02-28T19:32:02.718555Z"Osipov, D. V."https://zbmath.org/authors/?q=ai:osipov.denis-vLet \(A\) be a commutative ring and \(A((t))\) the algebra of Laurent series over \(A\). Using the concept of Contou-Carrère symbol (see [\textit{P. Deligne}, Publ. Math., Inst. Hautes Étud. Sci. 73, 147--181 (1991; Zbl 0749.14011)]), the author defines the formal Bott-Thurston cocycle as a certain 2-cocycle on the group of continuous \(A\)-automorphisms of \(A((t))\) taking values in the group of invertible elements of \(A\). The main result of the paper under review is a formal version of the Riemann-Roch theorem,
applicable to separated schemes over \(\mathbb Q\).
The proof is partly based on ideas and constructions similar to those discussed in [\textit{M. Kapranov} and \textit{É. Vasserot}, Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 113--133 (2007; Zbl 1129.14022)].
Reviewer: Aleksandr G. Aleksandrov (Moskva)Modular Koszul duality for Soergel bimoduleshttps://zbmath.org/1527.140372024-02-28T19:32:02.718555Z"Makisumi, Shotaro"https://zbmath.org/authors/?q=ai:makisumi.shotaroSummary: We generalize the modular Koszul duality of \textit{P. N. Achar} and \textit{S. Riche} [Duke Math. J. 165, No. 1, 161--215 (2016; Zbl 1375.14162)] to the setting of Soergel bimodules associated to any finite Coxeter system. The key new tools are a functorial monodromy action and wall-crossing functors in the mixed modular derived category of Achar and Riche [loc. cit.]. In characteristic 0, this duality together with Soergel's conjecture (proved by \textit{B. Elias} and \textit{G. Williamson} [Ann. Math. (2) 180, No. 3, 1089--1136 (2014; Zbl 1326.20005)]) imply that our Soergel-theoretic graded category \(\mathcal{O}\) is Koszul self-dual, generalizing the result of \textit{W. Soergel} [J. Am. Math. Soc. 3, No. 2, 421--445 (1990; Zbl 0747.17008)] and \textit{A. Beilinson} et al. [J. Am. Math. Soc. 9, No. 2, 473--527 (1996; Zbl 0864.17006)].Operations on the Hochschild bicomplex of a diagram of algebrashttps://zbmath.org/1527.160122024-02-28T19:32:02.718555Z"Hawkins, Eli"https://zbmath.org/authors/?q=ai:hawkins.eliIn this paper, the author constructs an operad acting on the Hochschild bicomplex of a diagram of algebras, which is a functor in a category of associative algebras. First, the author reviews the notion of operads and describe two known operads. Then, he obtains a new operad for diagram of algebras. Using this operad, he proves that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. This operad can also be adapted for the reduced and asimplicial subcomplexes. He also shows that the total complex is a \(L_{\infty}\)-algebra. This \(L_{\infty}\)-structure enables to obtain results similar to those related to the famous Kontsevich's formality map, which is a \(L_{\infty}\)-quasi isomorphism.
Reviewer: Angela Gammella-Mathieu (Metz)Specht property of varieties of graded Lie algebrashttps://zbmath.org/1527.160242024-02-28T19:32:02.718555Z"Martinez Correa, Daniela"https://zbmath.org/authors/?q=ai:martinez-correa.daniela"Koshlukov, Plamen"https://zbmath.org/authors/?q=ai:koshlukov.plamen-e\textit{A. R. Kemer}'s [Math. USSR, Izv. 25, 359--374 (1985; Zbl 0586.16010); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 1042--1059 (1984)] proof of the Specht conjecture for associative algebras in characteristic zero helped fuel the continuing interest in the question of which algebraic structures admit a finite basis for their identities. The well-written paper under review includes a nice description of this subject in the introduction.
As for the body of the paper, it focuses on the \(\mathbb Z_n\)-graded identities of \(UT_n(F)^{(-)}\), the Lie algebra of \(n\times n\) upper triangular matrices. There are two main results: If the characteristic is 0 or is greater than or equal to \(n\), then any \(\mathbb Z_n\)-graded Lie algebra satisfying all of the graded Lie identities of \(UT_n(F)^{(-)}\) has a finite basis of identities. On the other hand, if \(F\) is an infinite field of characteristic~2, then there is a graded Lie algebra satisfying all of the graded Lie identities of \(UT_3(F)^{(-)}\) whose ideal of identities is not finitely based.
Reviewer: Allan Berele (Chicago)Universal enveloping algebra of a pair of compatible Lie bracketshttps://zbmath.org/1527.160272024-02-28T19:32:02.718555Z"Gubarev, Vsevolod"https://zbmath.org/authors/?q=ai:gubarev.vsevolod-yurevichAn algebra \(\langle L, [.,.]_1, [.,.]_2,+\rangle\) belongs to a variety \(Lie_2\) of pairs of compatible Lie brackets if \(\alpha[.,.]_2+\beta[.,.]_2\) is a Lie bracket for all \(\alpha,\beta\in \Bbbk\). Examples of such algebras can be found in [\textit{I. Z. Golubchik} and \textit{V. V. Sokolov}, Funct. Anal. Appl. 36, No. 3, 172--181 (2002; Zbl 1022.17024); translation from Funkts. Anal. Prilozh. 36, No. 3, 9--19 (2002)].
In the paper under review ``using the Poincare-Birkhoff-Witt property and the Gröbner-Shirshov bases technique, the author finds the linear basis of the associative universal enveloping algebra in the sense of Ginzburg and Kapranov''. Also the growth rate of such algebras is investigated.
Reviewer: Dmitry Artamonov (Moskva)Tensor product of evolution algebrashttps://zbmath.org/1527.170012024-02-28T19:32:02.718555Z"Cabrera Casado, Yolanda"https://zbmath.org/authors/?q=ai:cabrera-casado.yolanda"Martín Barquero, Dolores"https://zbmath.org/authors/?q=ai:martin-barquero.dolores"Martín González, Cándido"https://zbmath.org/authors/?q=ai:martin-gonzalez.candido"Tocino, Alicia"https://zbmath.org/authors/?q=ai:tocino.aliciaThe paper introduces for commutative \(K\)-algebras, the definition of being locally nondegenerate and proves that two-dimensional perfect evolution \(K\)-algebras and two-dimensional evolution \(K\)-algebras with one-dimensional square are locally nondegenerate. Also shown that perfectness is transferred from the tensor product to the factors and conversely. By searching conditions that ensure that the property of being an evolution algebra is inherited from the tensor product to the factors and conversely, proved that for the given two evolution algebras tensor product of these algebras also an evolution \(K\)-algebra. Furthermore, tensor product of the natural bases of the given two algebras will be a natural basis of the tensor product of the given evolution algebras. Given an example of anticommutative algebras (not evolution algebras) whose tensor product is an evolution algebra. However, if the ground field is algebraically closed, any four-dimensional tensorially decomposable simple evolution algebra is the tensor product of (simple) evolution algebras. Also showed that in arbitrary dimension, that nondegeneracy is inherited from the tensor product to the factors and conversely.
By authors it is proved that when one of the two factors of the tensor product is perfect, has an ideal of codimension 1 and the tensor product is an evolution algebra, then the other factor is an evolution algebra.
Computing the number of zeros \(z\) (and of zeros in the diagonal \(zd)\) of the Kronecker product in terms of the corresponding numbers \(z\) and \(zd\) of the factors allowed screening the \(4\times 4\) matrices that arise as the Kronecker product of \(2\times 2\) matrices. Also given a classification of four-dimensional tensorially decomposable perfect evolution algebras into 13 classes and determined some complete sets of invariants relative to such classification, some of them based on characteristic and minimal polynomials.
Reviewer: Sherzod N. Murodov (Bukhārā)Classification of \(D\)-bialgebra structures on power series algebrashttps://zbmath.org/1527.170022024-02-28T19:32:02.718555Z"Abedin, Raschid"https://zbmath.org/authors/?q=ai:abedin.raschidThe author uses algebro-geometric methods to classify the D-bialgebra structures on the power series algebra \(A[[z]]\) for certain central simple non-associative algebras \(A.\) These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over \(A\). If \(A\) is a Lie algebra, the author obtains new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on \(A[[z]]\) as well as of non-degenerate solutions to the usual CYBE. If \(A\) is associative, the author classifies the non-triangular topological balanced infinitesimal bialgebra structures on \(A[[z]]\) as well as all non-degenerate solutions to an associative version of CYBE.
Reviewer: Alexandre P. Pojidaev (Novosibirsk)A note on optimal systems of certain low-dimensional Lie algebrashttps://zbmath.org/1527.170032024-02-28T19:32:02.718555Z"Singh, Manjit"https://zbmath.org/authors/?q=ai:singh.manjit.1|singh.manjit"Gupta, Rajesh Kumar"https://zbmath.org/authors/?q=ai:gupta.rajesh-kumar.1|gupta.rajesh-kumarSummary: Optimal classifications of Lie algebras of some well-known equations under their group of inner automorphism are re-considered. By writing vector fields of some known Lie algebras in the abstract format, we have proved that there exist explicit isomorphism between Lie algebras and subalgebras which have already been classified. The isomorphism between Lie algebras is useful in the sense that the classifications of sub-algebras of dimension \(\leq 4\) have previously been carried out in literature. These already available classifications can be used to write classification of any Lie algebra of dimension \(\leq 4\). As an example, the explicit isomorphism between Lie algebra of variant Boussinesq system and subalgebra \({A}_{3,5}^{1/2}\) is proved, and subsequently, optimal subalgebras up to dimension four are obtained. Besides this, some other examples of Lie algebras are also considered for explicit isomorphism.Determinantal ideals and the canonical commutation relations: classically or quantizedhttps://zbmath.org/1527.170042024-02-28T19:32:02.718555Z"Jakobsen, Hans Plesner"https://zbmath.org/authors/?q=ai:jakobsen.hans-plesnerSummary: We construct homomorphic images of \(su(n,n)^{{\mathbb{C}}}\) in Weyl algebras \({{\mathcal{H}}}_{2nr} \). More precisely, and using the Bernstein filtration of \({{\mathcal{H}}}_{2nr}, su(n,n)^{{\mathbb{C}}}\) is mapped into degree 2 elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of \({{\mathcal{H}}}_{2nr} \), these homomorphisms give all unitary highest weight representations of \(su(n,n)^{{\mathbb{C}}}\) thus reconstructing the Kashiwara-Vergne List for the Segal-Shale-Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of \(u(r)^{{\mathbb{C}}}\) into \({{\mathcal{H}}}_{2nr}\) whose image commutes with the image of \(su(n,n)^{{\mathbb{C}}} \), and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of \((r+1)\times (r+1)\) minors is prime. Here, of course, \(r\le n-1\) and for a fixed such \(r\), the space of any irreducible representation of \(su(n,n)^{{\mathbb{C}}}\) is annihilated by this ideal. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for \({{\mathcal{U}}}_q(su(n,n)^{{\mathbb{C}}})\). The Weyl Algebras are replaced by the Hayashi-Weyl algebras \({{\mathcal{H}}}{{\mathcal{W}}}_{2nr}\) and the Fock space by a \(q\)-Fock space. Further, determinants are replaced by \(q\)-determinants, and a homomorphism of \({{\mathcal{U}}}_q(u(r)^{{\mathbb{C}}})\) into \({{\mathcal{H}}}{{\mathcal{W}}}_{2nr}\) is constructed with analogous properties. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.Combinatorial and geometric constructions associated with the Kostant cascadehttps://zbmath.org/1527.170052024-02-28T19:32:02.718555Z"Panyushev, Dmitri I."https://zbmath.org/authors/?q=ai:panyushev.dmitri-iTake a simple algebraic group \(G\) and fix a triangular decomposition os it's Lie algebra \(g\). Take the corresponding root system. The Kostant cascade is a set \(\mathcal{K}\) of strongly orthogonal roots in the set \(\Delta_+\) of positive roots that is constructed recursively starting with the highest root \(\theta\in \Delta_+\). As the author writes ``over the years, I gathered a number of results related to the occurrences of \(\mathcal{K}\) in various problems of Combinatorics, Invariant Theory, and Representation Theory.'' Some of these observations appear in the present article. In particular the author discusses properties of certain objects naturally associated with \(\mathcal{K}\): an abelian ideal of the Borel subalgebra \(\mathfrak{b}\), a nilpotent \(G\)-orbit in \(g\), and an involution of \(g\).
Reviewer: Dmitry Artamonov (Moskva)Unrestricted quantum moduli algebras. I: The case of punctured sphereshttps://zbmath.org/1527.170062024-02-28T19:32:02.718555Z"Baseilhac, Stéphane"https://zbmath.org/authors/?q=ai:baseilhac.stephane"Roche, Philippe"https://zbmath.org/authors/?q=ai:roche.philippe|roche.philippe-eSummary: Let \(\Sigma\) be a finite type surface, and \(G\) a complex algebraic simple Lie group with Lie algebra \(\mathfrak{g}\). The quantum moduli algebra of \((\Sigma,G)\) is a quantization of the ring of functions of \(X_G(\Sigma)\), the variety of \(G\)-characters of \(\pi_1(\Sigma)\), introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid 1990s. It can be realized as the invariant subalgebra of so-called graph algebras, which are \(U_q(\mathfrak{g})\)-module-algebras associated to graphs on \(\Sigma\), where \(U_q(\mathfrak{g})\) is the quantum group corresponding to \(G\). We study the structure of the quantum moduli algebra in the case where \(\Sigma\) is a sphere with \(n+1\) open disks removed, \(n\geq 1\), using the graph algebra of the ``daisy'' graph on \(\Sigma\) to make computations easier. We provide new results that hold for arbitrary \(G\) and generic \(q\), and develop the theory in the case where \(q=\epsilon\), a primitive root of unity of odd order, and \(G=\mathrm{SL}(2,\mathbb{C})\). In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring \(\mathcal{O}(G^n)\). We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on \(\mathcal{O}(G^n)\). We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of \(\mathbb{C}[X_G(\Sigma)]\) endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra \(K_{\zeta}(\Sigma)\) at \(\zeta:=\mathrm{i}\epsilon^{1/2}\) with this quantum moduli algebra specialized at \(q=\epsilon\). This allows us to recast in the quantum moduli setup some recent results of Bonahon-Wong and Frohman-Kania-Bartoszyńska-Lê on \(K_{\zeta}(\Sigma)\).A realization of the enveloping superalgebra \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\)https://zbmath.org/1527.170072024-02-28T19:32:02.718555Z"Du, Jie"https://zbmath.org/authors/?q=ai:du.jie"Fu, Qiang"https://zbmath.org/authors/?q=ai:fu.qiang"Lin, Yanan"https://zbmath.org/authors/?q=ai:lin.yananAuthors' abstract: ``In [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)], \textit{A. A. Beilinson} et al. (BLM) gave a beautiful realization for quantum \(\mathfrak{gl}_n\) via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\) of the loop algebra \(\widehat{\mathfrak{gl}}_{m|n}\) of \({\mathfrak{gl}}_{m|n}\) with those of affine symmetric groups \(\widehat{\mathfrak{S}}_r\). Then, we give a BLM type realization of \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\) via affine Schur superalgebras.
The first application of the realization of \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\) is to determine the action of \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\) on tensor spaces of the natural representation of \(\widehat{\mathfrak{gl}}_{m|n}\). These results in epimorphisms from \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\) to affine Schur superalgebras so that the bridging relation between representations of \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\) and \(\widehat{\mathfrak{S}}_r\) is established. As a second application, we construct a Kostant type \(\mathbb{Z}\)-form for \(\mathcal{U}_{\mathbb{Q}}(\widehat{\mathfrak{gl}}_{m|n})\) whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur-Weyl duality in arbitrary characteristics.''
The constructions involve many iterative computations.
Reviewer: Wilberd van der Kallen (Utrecht)Quantum cluster characters of Hall algebras revisitedhttps://zbmath.org/1527.170082024-02-28T19:32:02.718555Z"Fu, Changjian"https://zbmath.org/authors/?q=ai:fu.changjian"Peng, Liangang"https://zbmath.org/authors/?q=ai:peng.liangang"Zhang, Haicheng"https://zbmath.org/authors/?q=ai:zhang.haichengSummary: Let \(Q\) be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of \(Q\). As an application, we recover the surjective homomorphism defined in [\textit{Ming Ding}, \textit{Fan Xu} and the third author, Math. Z. 296, No. 3-4, 945--968 (2020; Zbl 1509.17010)], which realizes the principal coefficient quantum cluster algebra \(\mathcal{A}_q(Q)\) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero-Chapoton formula, as well as some multiplication formulas between quantum Caldero-Chapoton characters.Regular representations of quantum supergroups \(U_{\upsilon}(\mathfrak{g} \mathfrak{l}_{m|n})\) and \(U_{\upsilon}(\mathfrak{q}_{n})\)https://zbmath.org/1527.170092024-02-28T19:32:02.718555Z"Gu, Haixia"https://zbmath.org/authors/?q=ai:gu.haixia"Zhou, Zhongguo"https://zbmath.org/authors/?q=ai:zhou.zhongguoThe Lie superalgebras \(\mathfrak{gl}_{m|n}\) and \(\mathfrak{q}_{n}\) over \(\mathbb{C}\) are two superalgebras associated with the simple superalgebras \(M_{m|n}(\mathbb{C})\) and \(Q_{n}(\mathbb{C})\), respectively.
In the paper under review, the authors have collected the descriptions of the regular representations of quantum supergroups \(U_{v}(\mathfrak{gl}_{m|n})\) and \(U_{v}(\mathfrak{q}_{n})\). \textit{A. A. Beilinson} et al. in [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)] achieved the realization of quantum group \(U_{v}(\widehat{\mathfrak{gl}}_{m|n})\) (now called BLM realization). This paper includes a BLM reconstruction of \(U_{v}(\mathfrak{gl}_{m|n})\) through \(v\)-Schur superalgebra, and a reconstruction of the regular representation of \(U_{v}(\mathfrak{gl}_{m|n})\) via super symmetric algebra associated with the natural representation. Generalizing the second method to \(U_{v}(\mathfrak{q}_{n})\), the authors depict the regular representation of \(U_{v}(\mathfrak{q}_{n})\) via differential operators over a deformed polynomial superalgebra.
For the entire collection see [Zbl 1508.20002].
Reviewer: Egle Bettio (Venezia)Polyhedral realizations for \(B(\infty)\) and extended Young diagrams, Young walls of type \(\mathrm{A}^{(1)}_{n-1}\), \(\mathrm{C}^{(1)}_{n-1}\), \(\mathrm{A}^{(2)}_{2n-2}\), \(\mathrm{D}^{(2)}_n\)https://zbmath.org/1527.170102024-02-28T19:32:02.718555Z"Kanakubo, Yuki"https://zbmath.org/authors/?q=ai:kanakubo.yukiSummary: The crystal bases are quite useful combinatorial tools to study the representations of quantized universal enveloping algebras \(U_q(\mathfrak{g})\). The polyhedral realization for \(B(\infty)\) is a combinatorial description of the crystal base, which is defined as an image of embedding \({\Psi}_{\iota}:B(\infty)\hookrightarrow \mathbb{Z}^{\infty}_{\iota}\), where \(\iota\) is an infinite sequence of indices and \(\mathbb{Z}^{\infty}_{\iota}\) is an infinite \(\mathbb{Z}\)-lattice with a crystal structure associated with \(\iota\). It is a natural problem to find an explicit form of the polyhedral realization Im \((\Psi_\iota)\). In this article, supposing that \(\mathfrak{g}\) is of affine type \(\mathrm{A}^{(1)}_{n-1}\), \(\mathrm{C}^{(1)}_{n-1}\), \(\mathrm{A}^{(2)}_{2n-2}\) or \(\mathrm{D}^{(2)}_n\) and \(\iota\) satisfies the condition of `adaptedness', we describe Im \((\Psi_\iota)\) by using several combinatorial objects such as extended Young diagrams and Young walls.Nilpotency of Lie type algebras with metacyclic Frobenius groups of automorphismshttps://zbmath.org/1527.170112024-02-28T19:32:02.718555Z"Makarenko, N. Yu."https://zbmath.org/authors/?q=ai:makarenko.n-yuIn the article, the author studies algebras of Lie type (not necessarily finite-dimensional) admitting a finite Frobenius group as a subgroup of the automorphism group. The following statement was proven.
Assume that a Lie-type algebra admits a finite Frobenius group of automorphisms with cyclic kernel \(F\) of order \(n\) and complement \(H\) of order \(q\) such that the fixed-point subalgebra with respect to \(F\) is trivial and the fixed-point subalgebra with respect to \(H\) is nilpotent of class \(c\). If the ground field contains a primitive \(n\)-th root of unity, then the algebra is nilpotent and the nilpotency class is bounded in terms of \(q\) and \(c\).
Reviewer: Ilya Gorshkov (Novosibirsk)Correction to: ``Lie algebras graded by the weight systems \((\Theta_3,\mathrm{sl}_3)\) and \((\Theta_4,\mathrm{sl}_4)\)''https://zbmath.org/1527.170122024-02-28T19:32:02.718555Z"Yaseen, Hogir Mohammed"https://zbmath.org/authors/?q=ai:yaseen.hogir-mohammedCorrection to the author's paper [ibid. 8, 765--783 (2022; Zbl 1523.17043)].Representations of the BMS-Kac-Moody algebrahttps://zbmath.org/1527.170132024-02-28T19:32:02.718555Z"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu|li.jinlu.1"Sun, Jiancai"https://zbmath.org/authors/?q=ai:sun.jiancaiSummary: We investigate the BMS-Kac-Moody algebra \(\mathcal{L}\) with two \(u(1)\) Kac-Moody currents without central extensions. We give a construction of simple restricted modules generalizing both highest weight modules and Whittaker modules for \(\mathcal{L}\). More precisely, we establish a 1-1 correspondence between simple restricted \(\mathcal{L}\)-modules and simple modules of a family of finite-dimensional solvable Lie algebras associated to \(\mathcal{L}\). Also, we present several equivalent descriptions for simple restricted modules over \(\mathcal{L}\).Sewn sphere cohomologies for vertex algebras.https://zbmath.org/1527.170142024-02-28T19:32:02.718555Z"Zuevsky, Alexander"https://zbmath.org/authors/?q=ai:zuevsky.alexanderSummary: We define sewn elliptic cohomologies for vertex algebras by sewing procedure for coboundary operators.From Lie algebra crossed modules to tensor hierarchieshttps://zbmath.org/1527.170152024-02-28T19:32:02.718555Z"Lavau, Sylvain"https://zbmath.org/authors/?q=ai:lavau.sylvain"Stasheff, Jim"https://zbmath.org/authors/?q=ai:stasheff.james-dThe present paper establishes the mathematical credentials of tensor hierarchies, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing -- the embedding tensor -- between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. The authors show that any Lie-Leibniz triple induces a differential graded Lie algebra -- its associated tensor hierarchy -- whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ``Lie-ization'' of the former.
Reviewer: Alexandre P. Pojidaev (Novosibirsk)A new kind of soft algebraic structures: bipolar soft Lie algebrashttps://zbmath.org/1527.170162024-02-28T19:32:02.718555Z"Çıtak, F."https://zbmath.org/authors/?q=ai:citak.filizSummary: In this paper, basic concepts of soft set theory was mentioned. Then, bipolar soft Lie algebras and bipolar soft Lie ideals were defined with the help of soft sets. Some algebraic properties of the new concepts were investigated. The relationship between the two structures were analyzed. Also, it was proved that the level cuts of a bipolar soft Lie algebra were Lie subalgebras of a Lie algebra by the new definitions. After then, soft image and soft preimage of a bipolar soft Lie algebra/ideal were proved to be a bipolar soft Lie algebra/ideal.Albert algebras over \(\mathbb{Z}\) and other ringshttps://zbmath.org/1527.170172024-02-28T19:32:02.718555Z"Garibaldi, Skip"https://zbmath.org/authors/?q=ai:garibaldi.skip"Petersson, Holger P."https://zbmath.org/authors/?q=ai:petersson.holger-p"Racine, Michel L."https://zbmath.org/authors/?q=ai:racine.michel-lThe paper studies Jordan algebras. The authors are interested in the Albert algebras. The classification of the simple Jordan algebras is well known, and the Albert algebra is the only one of these which is exceptional (that is it does not come from an associative algebra by means of passing to the symmetric, Jordan product). Moreover Albert algebras are closely related to simple affine group schemes of type \(F_4\), \(E_6\), \(E_7\). The authors of the paper under review classify Albert algebras over the ring of the integers \(\mathbb{Z}\). In fact they reduce the classification to that of groups of type \(F_4\) given by \textit{B. Conrad} [Panor. Synth. 46, 193--253 (2015; Zbl 1356.14033)]. They also describe various aspects of the structure of Albert algebras over semilocal rings. The authors take great care in describing the novelty of their results, and the differences in their approach to the study of Albert algebras compared to the classical ones. The paper is well written and will be of significant interest to people working on nonassociative algebras.
Reviewer: Plamen Koshlukov (Campinas)\(\ast\)-Lie-type maps on alternative \(\ast\)-algebrashttps://zbmath.org/1527.170182024-02-28T19:32:02.718555Z"de Oliveira Andrade, Aline Jaqueline"https://zbmath.org/authors/?q=ai:de-oliveira-andrade.aline-jaqueline"Barreiro, Elisabete"https://zbmath.org/authors/?q=ai:barreiro.elisabete"Macedo Ferreira, Bruno Leonardo"https://zbmath.org/authors/?q=ai:ferreira.bruno-leonardo-macedoLet \(\mathcal{A}\) be the class of alternative unital \(*\)-algebras, where \(*\) is an involution. The authors obtain some sufficient conditions for multiplicative \(*\)-Lie-type maps on algebras in \(\mathcal{A}\) to be \(*\)-additive or to be \(*\)-isomorphism. As an application, a result on alternative \(W^*\)-algebras is obtained.
Reviewer: Alexandre P. Pojidaev (Novosibirsk)Embedding of Malcev and alternative algebrashttps://zbmath.org/1527.170192024-02-28T19:32:02.718555Z"Kornev, Alexandr I."https://zbmath.org/authors/?q=ai:kornev.aleksandr-iThe author introduces the notions of \(g\)-associative and \(g\)-Lie algebras as a generalization of associative and Lie algebras, respectively, where \(g(x,y)\) is a bilinear polynomial in a free algebra with involution. The author proves that every alternative (Malcev) algebra is embedded into a \(g\)-associative (\(g\)-Lie) algebra and shows that every Malcev algebra is embedded as a commutator subalgebra into a \(g\)-associative algebra in two different ways (for two different polynomials \(g(x, y)\)).
Reviewer: Alexandre P. Pojidaev (Novosibirsk)Markov evolution algebrashttps://zbmath.org/1527.170202024-02-28T19:32:02.718555Z"Paniello, Irene"https://zbmath.org/authors/?q=ai:paniello.ireneThe paper is devoted to study the Markov evolution algebras, that is, evolution algebras having Markov structure matrices. By author considered the discrete-time case, and delve into their algebraic structure for later application to continuous-time Markov evolution algebras that arise defined by standard stochastic semigroups.
Proved that for each homogeneous Markov chain there is an evolution algebra whose structural constants are transition probabilities, and whose generator set is the state space of the Markov chain. Also given beautiful examples.
Showed that for the given Markov evolution algebra, if it is a baric algebra, then the underlying homogeneous discrete-time (HDT) Markov chain is reducible. Also proved that the Markov evolution algebras are not nil (nor nilpotent) and have no nonzero absolute nilpotent elements. Given conditions for the evolution element of the Markov evolution algebra to be idempotent.
Author proved that Markov evolution algebras are nondegenerate. Also gives the condition for the Markov evolution algebras to be indecomposable.
Author exploits some well-known Markov processes (Birth and death processes, Greenwood model, Jukes-Cantor model) modeling different stochastic processes to define the corresponding Markov evolution algebras.
With the help of the continuous-time Markov chains author studied continuous-time Markov evolution algebras and gives the condition for the continuous-time Markov evolution algebras to be basic simple.
Reviewer: Sherzod N. Murodov (Bukhārā)Inverse \(K\)-Chevalley formulas for semi-infinite flag manifolds. I: Minuscule weights in ADE typehttps://zbmath.org/1527.200032024-02-28T19:32:02.718555Z"Kouno, Takafumi"https://zbmath.org/authors/?q=ai:kouno.takafumi"Naito, Satoshi"https://zbmath.org/authors/?q=ai:naito.satoshi"Orr, Daniel"https://zbmath.org/authors/?q=ai:orr.daniel.1"Sagaki, Daisuke"https://zbmath.org/authors/?q=ai:sagaki.daisukeSummary: We prove an explicit inverse Chevalley formula in the equivariant \(K\)-theory of semi-infinite flag manifolds of simply laced type. By an `inverse Chevalley formula' we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a \(\mathbb{Z}\left [q^{\pm 1}\right]\)-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars \(e^{\lambda}\), where \(\lambda\) is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type \(E_8\). The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric \(q\)-Toda operators for minuscule weights in ADE type.The classical Tits quadrangleshttps://zbmath.org/1527.200422024-02-28T19:32:02.718555Z"Mühlherr, Bernhard"https://zbmath.org/authors/?q=ai:muhlherr.bernhard-matthias"Weiss, Richard M."https://zbmath.org/authors/?q=ai:weiss.richard-mMoufang polygons were classified by \textit{J. Tits} and the second author [Moufang polygons. Berlin: Springer (2002; Zbl 1010.20017)]. Tits polygons, as defined by the authors and \textit{H. P. Petersson} [Tits polygons. Providence, RI: American Mathematical Society (2022; Zbl 1498.51001)] generalize Moufang polygons. The authors, in a series of papers classified the dagger-sharp Tits triangles, hexagons and octagons that are \(7\)-plump.
In the paper under review, the authors prove that a genuine Tits quadrangle that is \(5\)-sturdy and laser-sharp but not exceptional is uniquely determined by either a quadratic space over a field or a pseudo-quadratic module over a simple associative ring with involution. This result completes the classification of \(7\)-sturdy laser-sharp Tits polygons.
Reviewer: Egle Bettio (Venezia)Rigid reflections of rank 3 Coxeter groups and reduced roots of rank 2 Kac-Moody algebrashttps://zbmath.org/1527.200592024-02-28T19:32:02.718555Z"Lee, Kyu-Hwan"https://zbmath.org/authors/?q=ai:lee.kyu-hwan"Yu, Jeongwoo"https://zbmath.org/authors/?q=ai:yu.jeongwooIn a recent paper by K.-H. Lee and K. Lee, rigid reflections are defined for any Coxeter group via non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, the rigid reflections are related to the rigid representations of the quiver. For a family of rank \(3\) Coxeter groups, it was conjectured in the same paper that there is a natural bijection from the set of reduced positive roots of a symmetric rank \(2\) Kac-Moody algebra onto the set of rigid reflections of the corresponding rank \(3\) Coxeter group. The authors prove this conjecture.
Reviewer: Erich W. Ellers (Toronto)Character sheaves for classical symmetric pairshttps://zbmath.org/1527.200772024-02-28T19:32:02.718555Z"Vilonen, Kari"https://zbmath.org/authors/?q=ai:vilonen.kari"Xue, Ting"https://zbmath.org/authors/?q=ai:xue.tingLet \( G \) be a connected reductive algebraic group over \(\mathbb{C}\) and \( K = G^{\theta} \) a symmetric subgroup fixed by an involution \( \theta \in \mathrm{Aut}\, G \). Denote by the same letter \( \theta \) its differential, which is an involution on the Lie algebra \( \mathfrak{g} \) of \( G \). Let \(\mathfrak{g} =\mathfrak{g}_0 +\mathfrak{g}_1 \) be the eigenspace decomposition of \( \theta \), where \(\mathfrak{g}_0 \) is the Lie algebra of \( K \) so that \(\mathfrak{g}_1 \simeq T_e (G/K) \). Let \(\mathcal{N}\subset\mathfrak{g} \) be the nilpotent variety and put \(\mathcal{N}_1 = \mathcal{N}\cap\mathfrak{g}_1 \).
A character sheaf for the symmetric pair \( (G,K) \) is an irreducible \(\mathbb{C}^*\)-conic \( K \)-equivariant perverse sheaf on \(\mathfrak{g}_1 \) whose singular support is contained in \(\mathfrak{g}_1 \times\mathcal{N}_1 \), where \(\mathfrak{g}_1 \times\mathcal{N}_1 \) is considered to be a subset in \( T^*\mathfrak{g}_1 \). The set of character sheaves is denoted by \(\mathrm{Char}_K(\mathfrak{g}_1)\). Note that \(\mathrm{Char}_K(\mathfrak{g}_1\) is in bijection with the set of irreducible \( K \)-equivariant perverse sheaves on \(\mathcal{N}_1 \) via Fourier transform. The notion of the character sheaves for a symmetric pair is a generalization of those defined by \textit{G. Lusztig} for \( \mathfrak{g} \) [Invent. Math. 75, 205--272 (1984; Zbl 0547.20032)].
In this paper under review, the authors classify all character sheaves for classical symmetric pairs (Theorems 6.2--6.4).
There are some important classes of character sheaves, namely (i) full support character sheaves denoted by \(\mathrm{Char}_K(\mathfrak{g}_1)\), whose support is all of \(\mathfrak{g}_1 \); (ii) nilpotent support character sheaves \(\mathrm{Char}_K^{\mathrm{n}}(\mathfrak{g}_1)\) supported in the closure of a nilpotent orbit; (iii) cuspidal character sheaves \(\mathrm{Char}_K^{\mathrm{cusp}}(\mathfrak{g}_1)\).
(i) The full support character sheaves are obtained as a \( K \)-equivariant IC sheaves on \(\mathfrak{g}_1^{\mathrm{rs}} \), the set of regular semisimple elements, and local systems on it. These IC sheaves are obtained from the Fourier transform of nearby cycle perverse sheaves on the nilpotent cone [\textit{M. Grinberg} et al., Am. J. Math. 145, No. 1, 1--63 (2023; Zbl 07653690)] and explicitly described by the representations of Hecke algebras (with certain unequal parameters, see [\textit{R. Dipper} and \textit{G. James}, J. Algebra 146, No. 2, 454--481 (1992; Zbl 0808.20016); \textit{S. Ariki} and \textit{A. Mathas}, Math. Z. 233, No. 3, 601--623 (2000; Zbl 0955.20003)] and additional \( 2 \)-groups called R-group (Corollary 5.5).
(ii) The nilpotent support character sheaves are supported on the Richardson orbits [\textit{P. E. Trapa}, J. Algebra 286, No. 2, 361--385 (2005; Zbl 1070.22003)] and classified in Theorems 4.2 and 4.4. They only appear in the case where \( \theta \) is inner (equivalently, when there exist \( \theta \)-stable Borel subgroups).
(iii) The most fundamental ones are cuspidal character sheaves and the rest of the character sheaves appear as a direct summand of parabolic inductions from cuspidal ones. They are classified in Theorem 1.1 (Corollary 6.7).
Reviewer: Kyo Nishiyama (Aoyama)Degenerate quantum general linear groupshttps://zbmath.org/1527.200822024-02-28T19:32:02.718555Z"Cheng, Jin"https://zbmath.org/authors/?q=ai:cheng.jin.1"Wang, Yan"https://zbmath.org/authors/?q=ai:wang.yan.192"Zhang, Ruibin"https://zbmath.org/authors/?q=ai:zhang.ruibinSummary: Given any pair of positive integers \(m\) and \(n\), we construct a new Hopf algebra, which may be regarded as a degenerate version of the quantum group of \(\mathfrak{gl}_{m+n}\). We study its structure and develop a highest weight representation theory. The finite dimensional simple modules are classified in terms of highest weights, which are essentially characterised by \(m + n - 2\) nonnegative integers and two arbitrary nonzero scalars. In the special case with \(m = 2\) and \(n = 1\), an explicit basis is constructed for each finite dimensional simple module. For all \(m\) and \(n\), the degenerate quantum group has a natural irreducible representation acting on \(\mathbb{C}(q)^{m+n}\). It admits an \(R\)-matrix that satisfies the Yang-Baxter equation and intertwines the co-multiplication and its opposite. This in particular gives rise to isomorphisms between the two module structures of any tensor power of \(\mathbb{C}(q)^{m+n}\) defined relative to the co-multiplication and its opposite respectively. A topological invariant of knots is constructed from this \(R\)-matrix, which reproduces the celebrated HOMFLY polynomial. Degenerate quantum groups of other classical types are briefly discussed.The \(q\)-Schur algebras and \(q\)-Schur dualities of finite typehttps://zbmath.org/1527.200832024-02-28T19:32:02.718555Z"Luo, Li"https://zbmath.org/authors/?q=ai:luo.li.1|luo.li"Wang, Weiqiang"https://zbmath.org/authors/?q=ai:wang.weiqiang.3|wang.weiqiangSummary: We formulate a \(q\)-Schur algebra associated with an arbitrary \(W\)-invariant finite set \(X_{\text{f}}\) of integral weights for a complex simple Lie algebra with Weyl group \(W\). We establish a \(q\)-Schur duality between the \(q\)-Schur algebra and Hecke algebra associated with \(W\). We then realize geometrically the \(q\)-Schur algebra and duality and construct a canonical basis for the \(q\)-Schur algebra with positivity. With suitable choices of \(X_{\text{f}}\) in classical types, we recover the \(q\)-Schur algebras in the literature. Our \(q\)-Schur algebras are closely related to the category \({\mathcal{O}} \), where the type \(G_2\) is studied in detail.Slice quaternionic analysis in two variableshttps://zbmath.org/1527.300322024-02-28T19:32:02.718555Z"Dou, Xinyuan"https://zbmath.org/authors/?q=ai:dou.xinyuan"Ren, Guangbin"https://zbmath.org/authors/?q=ai:ren.guangbin"Sabadini, Irene"https://zbmath.org/authors/?q=ai:sabadini.irene"Wang, Xieping"https://zbmath.org/authors/?q=ai:wang.xiepingSummary: Slice quaternionic analysis in two variables is a generalization of the theory of several complex variables to quaternions. This study relies on the theory of stem functions and the theory of holomorphic functions in two complex variables. Our approach is to introduce holomorphicity for stem functions in terms of two commutative complex structures. It turns out that, locally, a function which is slice regular corresponds exactly to the Taylor series of two ordered quaternions, with the coefficients on the right. The Hartogs phenomenon holds in our setting; however, its proof is subtle due to some topological obstacles. We overcome them by showing that holomorphic stem functions preserve the property of being intrinsic after extension.Invariant solutions of the supersymmetric version of a two-phase fluid flow systemhttps://zbmath.org/1527.352812024-02-28T19:32:02.718555Z"Grundland, A. M."https://zbmath.org/authors/?q=ai:grundland.alfred-michel"Hariton, A. J."https://zbmath.org/authors/?q=ai:hariton.alexander-jSummary: A supersymmetric extension of a two-phase fluid flow system is formulated. A superalgebra of Lie symmetries of the supersymmetric extension of this system is computed. The classification of the one-dimensional subalgebras of this superalgebra into 63 equivalence classes is performed. For some of the subalgebras, it is found that the invariants have a non-standard structure. For six selected subalgebras, the symmetry reduction method is used to find invariants, orbits of the subgroups and reduced systems. Through the solutions of the reduced systems, the most general solutions are expressed in terms of arbitrary functions of one or two fermionic and one bosonic variables.Symmetries and conservation laws of the Liouville equationhttps://zbmath.org/1527.354532024-02-28T19:32:02.718555Z"Zharinov, V. V."https://zbmath.org/authors/?q=ai:zharinov.victor-vSummary: Symmetries and conservation laws of the Liouville equation are studied in the frames of the algebra-geometrical approach to partial differential equations.Integrable systems associated to the filtrations of Lie algebrashttps://zbmath.org/1527.370592024-02-28T19:32:02.718555Z"Jovanović, Božidar"https://zbmath.org/authors/?q=ai:jovanovic.bozidar-zarko|jovanovic.bozidar-d"Šukilović, Tijana"https://zbmath.org/authors/?q=ai:sukilovic.tijana"Vukmirović, Srdjan"https://zbmath.org/authors/?q=ai:vukmirovic.srdanThe paper is devoted to the study of integrable systems associated to the filtration of Lie algebras in the compact case. In particular, if \(G\) is a compact connected Lie group \(G\), the authors consider a chain of compact connected Lie subgroup \(G_0 \subset G_1 \subset \dots \subset G_{n-1} \subset G_n=G \) and the corresponding filtration of Lie algebras \({g}=\mathrm{Lie}(G)\), \({g}_0 \subset {g}_1 \subset \dots \subset {g}_{n-1} \subset {g}_n={g}\) and investigate integrable Euler equations associated to the filtration of Lie algebras. In the case \({g}_0=\{0\}\) and the natural filtration of \({so}(n)\) and \({u}(n)\) the so-called Gel'fand-Cetlin integrable systems are obtained. The authors prove complete integrability in a noncommutative sense (superintegrability) by means of polynomial integrals for filtrations of compact Lie algebras \({g}\), generalizing results known for Gel'fand-Cetlin systems. As remarked by the authors, superintegrability represent a stronger property than the usual commutative (or Liouville) integrability. The system is solvable by quadratures, regular compact invariant manifolds are isotropic tori, and there exist appropriate action-angle coordinates in which the dynamics is linearized. The paper concludes with several constructions of complete commutative polynomial integrals for the system.
Reviewer: Danilo Latini (Roma)Conversations with Flaschka: Kac-Moody groups and Verblunsky coefficientshttps://zbmath.org/1527.370602024-02-28T19:32:02.718555Z"Latifi, Mohammad Javad"https://zbmath.org/authors/?q=ai:latifi.mohammad-javad"Pickrell, Doug"https://zbmath.org/authors/?q=ai:pickrell.dougSummary: In this paper two items are discussed which in one way or another originated from conversations with Hermann Flaschka and his students. The first is an application of the Toda lattice to the question of whether there exists a complex Lie group with an exponential map associated to an indefinite type Kac-Moody Lie algebra. The second concerns a new example of the Verblunsky correspondence.Lie algebras and integrable systems: elastic curves and rolling geodesicshttps://zbmath.org/1527.370652024-02-28T19:32:02.718555Z"Jurdjevic, V."https://zbmath.org/authors/?q=ai:jurdjevic.velimirSummary: This paper follows a long-standing fascination in the relevance of Lie algebras and Lie groups for problems of applied mathematics. It originates with the discovery that the mathematical formalism initiated by G. Kirchhoff to model the equilibrium configurations of an elastic rod can be extended to the isometry groups of certain Riemannian manifolds through control theoretic insights and the Maximum Principle, giving rise to a large class of Hamiltonian systems that link geometry with physics in novel ways. This paper focuses on the relations between the Kirchhoff-like affine-quadratic problem and the rolling geodesic problem associated with the rollings of homogeneous manifolds \(G/K\), equipped with a \(G\)-invariant metric, on their tangent spaces. We will show that there is a remarkable connection between these two problems manifested through a common isospectral curve in the Lie algebra \(\mathfrak{g}\) of \(G\). In the process we will reveal the significance of curvature for the theory of elastic curves.Homogeneous spaces with invariant Koszul-Vinberg structureshttps://zbmath.org/1527.530492024-02-28T19:32:02.718555Z"Abouqateb, Abdelhak"https://zbmath.org/authors/?q=ai:abouqateb.abdelhak"Boucetta, Mohamed"https://zbmath.org/authors/?q=ai:boucetta.mohamed"Bourzik, Charif"https://zbmath.org/authors/?q=ai:bourzik.charifThis article is primarily devoted to providing an algebraic characterization of \(G\)-invariant Koszul-Vinberg structures on \(G\)-homogeneous manifolds. It relies to the spirit of Nomizu's theorem on invariant connections, as referenced in [\textit{K. Nomizu}, Am. J. Math. 76, 33--65 (1954; Zbl 0059.15805); \textit{A. Elduque}, Commun. Math. 28, No. 2, 199--229 (2020; Zbl 07300190)]. The authors give a list of examples, thus providing invariant Koszul-Vinberg structures on both reductive pairs and symmetric spaces. Besides, the article presents several examples of circumstances where the Koszul-Vinberg structures are pseudo-Hessian. A significant result suggests that if the Lie group \(G\) is semi-simple, \(G/H\) will not carry any non-trivial \(G\)-invariant pseudo-Hessian structure. This is explicitly formulated in Theorem 5.7. As a direct consequence of this result, the authors obtain a new proof of a result by \textit{H. Shima} [The geometry of Hessian structures. Hackensack, NJ: World Scientific (2007; Zbl 1244.53004), Theorem 9.2]. Another remarkable result of the paper reveals that the leaves of the affine foliation, which are linked to an invariant Koszul-Vinberg structure, are homogeneous pseudo-Hessian manifolds.
Reviewer: Zhuo Chen (Beijing)Poisson cohomology of 3D Lie algebrashttps://zbmath.org/1527.530692024-02-28T19:32:02.718555Z"Hoekstra, Douwe"https://zbmath.org/authors/?q=ai:hoekstra.douwe"Zeiser, Florian"https://zbmath.org/authors/?q=ai:zeiser.florianIntroduced by \textit{A. Lichnerowicz} in [J. Differ. Geom. 12, 253--300 (1977; Zbl 0405.53024)], Poisson cohomology stands as a fundamental subject in the realm of algebra. This paper makes a significant stride towards decoding this aspect, artfully computing the Poisson cohomology corresponding to specific cases of 3-dimensional Lie algebras. The findings drawn from this path-breaking work, when synthesized with those accrued by plugins such as [\textit{V. L. Ginzburg} and \textit{A. Weinstein}, J. Am. Math. Soc. 5, No. 2, 445--453 (1992; Zbl 0766.58018)] and [\textit{I. Mărcuţ} and \textit{F. Zeiser}, ``The Poisson cohomology of \(\mathfrak{s} \mathfrak{l}_2^\ast(\mathbb{R})\)'', Preprint, \url{arXiv:1911.11732}], formulate an encompassing spectrum of Poisson cohomology linked with all 3-dimensional Lie algebras. The methods harnessed in this investigation will yield robust influence and are ripe for expansion into a dynamic and more methodical theory. There is substantial hope that this theory could present feasible solutions for a multitude of complex challenges broadly encountered across the fields of cohomology and homology.
Reviewer: Zhuo Chen (Beijing)On odd parameters in geometryhttps://zbmath.org/1527.580022024-02-28T19:32:02.718555Z"Leites, Dimitry"https://zbmath.org/authors/?q=ai:leites.dimitry-aThis paper considers two types of occurrences of odd parameters:
\begin{itemize}
\item[1.] It considers deformations of several simple Lie superalgebras, giving a complete classification of deformations of simple finite-dimensional Lie superalgebras over \(\mathbb{C}\)\ with proofs for the first time. The author gives the adequate definition of Lie superalgebras and deformations with odd parameters. Some instances of odd parameters can be seen in [\textit{D. Leites}, in: Operator methods in ordinary and partial differential equations. Proceedings of the S. Kovalevsky symposium, Stockholm, Sweden, June 2000. Basel: Birkhäuser. 267--285 (2002; Zbl 1033.37037); \textit{D. A. Leites}, Theor. Math. Phys. 198, No. 2, 271--283 (2019; Zbl 1429.58006); translation from Teor. Mat. Fiz. 198, No. 2, 309--325 (2019); \textit{V. P. Palamodov}, Transl., Ser. 2, Am. Math. Soc. 175, 177--189 (1996; Zbl 0866.58014); \textit{V. N. Shander}, Funct. Anal. Appl. 26, No. 1, 55--56 (1992; Zbl 0832.17003); translation from Funkts. Anal. Prilozh. 26, No. 1, 69--71 (1992); \textit{K. Bettadapura}, Lett. Math. Phys. 109, No. 2, 381--402 (2019; Zbl 1412.32008); \textit{K. Bettadapura}, Doc. Math. 25, 65--91 (2020; Zbl 1441.14171)].
\item[2.] \textit{K. Gawedzki} [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 27, 335--366 (1978; Zbl 0369.53061)] and \textit{M. Batchelor} [Trans. Am. Math. Soc. 253, 329--338 (1979; Zbl 0413.58002)] showed that every smooth supermanifold is split. \textit{P. Green} [Proc. Am. Math. Soc. 85, 587--590 (1982; Zbl 0509.58001)] and \textit{V. P. Palamodov} [Funct. Anal. Appl. 17, 68--69 (1983; Zbl 0574.32042); translation from Funkts. Anal. Prilozh. 17, No. 1, 83--84 (1983)] established that a complex-analytic supermanifold can be non-split. So far, researchers considered only even obstructions to splitness. This paper shows that there are non-split supermanifolds of superdimension \(m\mid1\), the obstruction to their splitness depending on odd parameters.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Teleportation of masked informationhttps://zbmath.org/1527.810262024-02-28T19:32:02.718555Z"Abdelwahab, A. G."https://zbmath.org/authors/?q=ai:abdelwahab.a-g"Ghwail, S. A."https://zbmath.org/authors/?q=ai:ghwail.s-a"Metwally, N."https://zbmath.org/authors/?q=ai:metwally.nasser"Mahran, M. H."https://zbmath.org/authors/?q=ai:mahran.m-h"-S. F. Obada, A."https://zbmath.org/authors/?q=ai:s-f-obada.aSummary: In this contribution, we investigated the possibility of teleporting classical/quantum masked information, which may be coded either in a single qubit or qutrit. For this purpose, different systems are used as quantum channels; two-qubit, three-qubit, two qutrit systems, and different protocols are applied. All the teleported masked information are retrieved as masked states at the receiver station. The number of operations that may be performed by the receiver are limited. It is shown that, one can teleport masked classical information with maximum fidelity, while for quantum information the maximization depends on the weight parameter of the teleported state and the used joint measurements. Teleporting the total masked state is better than teleporting its marginals, where the fidelity of total masked state is maximum. However, the fidelity of teleporting masking quantum information via three qubit systems may be maximized by controlling the weight of the initial masked state and the polarization of the mediator. In some cases, the receiver need to diagonalize the final teleported state to maximize its fidelity, and consequently reduces the required local operations.Quantum particle on dual weight lattice in even Weyl alcovehttps://zbmath.org/1527.810712024-02-28T19:32:02.718555Z"Hrivnák, Jiří"https://zbmath.org/authors/?q=ai:hrivnak.jiri"Motlochová, Lenka"https://zbmath.org/authors/?q=ai:motlochova.lenka"Novotný, Petr"https://zbmath.org/authors/?q=ai:novotny.petrSummary: Even subgroups of affine Weyl groups corresponding to irreducible crystallographic root systems characterize families of single-particle quantum systems. Induced by primary and secondary sign homomorphisms of the Weyl groups, free propagations of the quantum particle on the refined dual weight lattices inside the rescaled even Weyl alcoves are determined by Hamiltonians of tight-binding types. Described by even hopping functions, amplitudes of the particle's jumps to the lattice neighbours are together with diverse boundary conditions incorporated through even hopping operators into the resulting even dual-weight Hamiltonians. Expressing the eigenenergies via weighted sums of the even Weyl orbit functions, the associated time-independent Schrödinger equations are exactly solved by applying the discrete even Fourier-Weyl transforms. Matrices of the even Hamiltonians together with specifications of the complementary boundary conditions are detailed for the \(C_2\) and \(G_2\) even dual-weight models.Shifted quantum groups and matter multiplets in supersymmetric gauge theorieshttps://zbmath.org/1527.810802024-02-28T19:32:02.718555Z"Bourgine, Jean-Emile"https://zbmath.org/authors/?q=ai:bourgine.jean-emileSummary: The notion of \textit{shifted} quantum groups has recently played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory of quantum groups. The first part of this paper presents new mathematical results for the shifted quantum toroidal \(\mathfrak{gl}(1)\) and quantum affine \(\mathfrak{sl}(2)\) algebras (resp. denoted \({\ddot{U}}_{q_1,q_2}^{\boldsymbol{\mu }}(\mathfrak{gl}(1))\) and \({\dot{U}}_q^{\boldsymbol{\mu }}(\mathfrak{sl}(2)))\). It defines several new representations, including finite dimensional highest \(\ell \)-weight representations for the toroidal algebra, and a vertex representation of \({\dot{U}}_q^{\boldsymbol{\mu }}(\mathfrak{sl}(2))\) acting on Hall-Littlewood polynomials. It also explores the relations between representations of \({\dot{U}}_q^{\boldsymbol{\mu }}(\mathfrak{sl}(2))\) and \({\ddot{U}}_{q_1,q_2}^{\boldsymbol{\mu }}(\mathfrak{gl}(1))\) in the limit \(q_1\rightarrow \infty\) (\(q_2\) fixed), and present the construction of several new intertwiners. These results are used in the second part to construct BPS observables for 5d \({{\mathcal{N}}}=1\) and 3d \({{\mathcal{N}}}=2\) gauge theories. In particular, it is shown that 5d hypermultiplets and 3d chiral multiplets can be introduced in the algebraic engineering framework using shifted representations, and the Higgsing procedure is revisited from this perspective.A generalization of a SIS epidemic model with fluctuationshttps://zbmath.org/1527.920492024-02-28T19:32:02.718555Z"Esen, Oğul"https://zbmath.org/authors/?q=ai:esen.ogul"Fernández-Saiz, Eduardo"https://zbmath.org/authors/?q=ai:fernandez-saiz.eduardo"Sardón, Cristina"https://zbmath.org/authors/?q=ai:sardon.cristina"Zając, Marcin"https://zbmath.org/authors/?q=ai:zajac.marcin(no abstract)