Let $\mathfrak g$ be a finite-dimensional Lie algebra. The symmetric algebra $\mathcal S(\mathfrak g)$ is equipped with the standard Lie-Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on $\mathcal S(\mathfrak g)$ to any finite order automorphism $θ$ of $\mathfrak g$. We study related Poisson-commutative subalgebras $\mathcal C$ of $\mathcal S(\mathfrak g)$ and associated Lie algebra contractions of $\mathfrak g$. To obtain substantial results, we have to assume that $\mathfrak g$ is semisimple. Then we can use Vinberg's theory of $θ$-groups and the machinery of Invariant Theory.
If $\mathfrak g=\mathfrak h\oplus\dots \oplus \mathfrak h$ (sum of $k$ copies), where $\mathfrak h$ is simple, and $θ$ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra $\mathcal C$ is polynomial and maximal. Furthermore, we quantise this $\mathcal C$ using a Gaudin subalgebra in the enveloping algebra $\mathcal U(\mathfrak g)$.
We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U(g) of a semisimple Lie algebra g. This family is parameterized by collections μ; z_1,...,z_n, where μ\in g^*, and z_1,...,z_n are pairwise distinct complex numbers. The construction presented here generalizes the famous construction of the higher Gaudin hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n=1, our construction gives a quantization of the family of maximal Poisson-commutative subalgebras in S(g) obtained by the argument shift method. Next, we des…
Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states…
We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra A=K[W] is said to be of Kostant type, if its centre Z(A) is freely generated by homogeneous polynomials F_1,...,F_r such that they give Kostant's regularity criterion on W (d_xF_i are linear independent if and only if the Poisson tensor has the maximal rank at x). If the initial Poisson algebra is of Kostant type and F_i satisfy a certain degree-equality, then the contraction is also of Kostant type. The general result is illustrated by two examples. Both are…
Let ${\mathcal S}(\mathfrak g)$ be the symmetric algebra of a reductive Lie algebra $\mathfrak g$ equipped with the standard Poisson structure. If ${\mathcal C}\subset\mathcal S(\mathfrak g)$ is a Poisson-commutative subalgebra, then ${\rm trdeg\,}{\mathcal C}\le\boldsymbol{b}(\mathfrak g)$, where $\boldsymbol{b}(\mathfrak g)=(\dim\mathfrak g+{\rm rk}\mathfrak g)/2$. We present a method for constructing the Poisson-commutative subalgebra $\mathcal Z_{\langle\mathfrak h,\mathfrak r\rangle}$ of transcendence degree $\boldsymbol{b}(\mathfrak g)$ via a vector space decomposition $\mathfrak g=\math…
We construct the Gerstenhaber bracket on Hochschild cohomology of a twisted tensor product of algebras, and, as examples, compute Gerstenhaber brackets for some quantum complete intersections arising in work of Buchweitz, Green, Madsen, and Solberg. We prove that a subalgebra of the Hochschild cohomology ring of a twisted tensor product, on which the twisting is trivial, is isomorphic, as Gerstenhaber algebras, to the tensor product of the respective subalgebras of the Hochschild cohomology rings of the factors.
Let $\mathfrak q$ be a Lie algebra over a field $\mathbb K$ and $p,\tilde p\in\mathbb K[t]$ two different normalised polynomials of degree at least 2. As vector spaces both quotient Lie algebras $\mathfrak q[t]/(p)$ and $\mathfrak q[t]/(\tilde p)$ can be identified with $W=\mathfrak q{\cdot}1\oplus\mathfrak q\bar t\oplus\ldots\oplus\mathfrak q\bar t^{n-1}$. If $\mathrm{deg}\,(p-\tilde p)$ is at most 1, then the Lie brackets $[\,\,,\,]_p$, $[\,\,,\,]_{\tilde p}$ induced on $W$ by $p$ and $\tilde p$, respectively, are compatible. By a general method, known as the Lenard-Magri scheme, we construc…
The symmetric algebra $S(\mathfrak g)$ of a reductive Lie algebra $\mathfrak g$ is equipped with the standard Poisson structure, i.e., the Lie-Poisson bracket. Poisson-commutative subalgebras of $S(\mathfrak g)$ attract a great deal of attention, because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset S(\mathfrak g)$ is bounded by the "magic number" $\boldsymbol{b}(\mathfrak g)$ of $\mathfrak g$. The "argument shift method" of Mishchenko-Fomenko was basically th…
A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent basis if and only if it is a nilpotent Lie algebra. When g is a semisimple Lie algebra, this allows us to define an ideal of S((g^n)^*)^G whose associated algebraic set in g^n is the set of n-tuples lying in a same Borel subalgebra.
For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie al…
We show that the space R^n x gl(n,R) with a certain antisymmetric bracket operation contains all n-dimensional Lie algebras. The bracket does not satisfy the Jacobi identity, but it does satisfy it for subalgebras which are isotropic under a certain symmetric bilinear form with values in R^n. We ask what the corresponding "group-like" object should be. The bracket may be obtained by linearizing at a point the bracket on TM + T*M introduced by T. Courant for the definition of Dirac structures, a notion which encompasses Poisson structures, closed 2-forms, and foliations.
Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf Aut}(\mathfrak g)$ a finite order automorphism, and $\mathfrak g_0$ the subalgebra of fixed points of $\vartheta$. Recently, we noticed that using $\vartheta$ one can construct a pencil of compatible Poisson brackets on $\mathcal S(\mathfrak g)$, and thereby a `large' Poisson-commutative subalgebra $\mathcal Z(\mathfrak g,\vartheta)$ of $\mathcal S(\mathfrak g)^{\mathfrak g_0}$. In this article, we study invariant-theoretic properties of $(\mathfrak g,\vartheta)$ that ensure good properties of $\mathcal Z(\mathfrak g,\vartheta)$…
We give the images of the adjoint representations of exceptional simple Lie algebras by matrices over complex numbers. Next, we digitalize these matrices by the use of Maxima, which is a computer algebra system. These digitalized matrices are provided by using Maxima's function. We prove that these digitalized matrices are closed by Lie bracket operations and make up simple Lie algebras. Moreover, to prove the type of the exceptional simple Lie algebra, we calculate the root system using Maxima for Lie bracket operations as matrix calculations. We show some examples of classical Lie subalgebra…
Explicit formulas for Segal-Sugawara vectors associated with the simple Lie algebra $\mathfrak{g}$ of type $G_2$ are found by using computer-assisted calculations. This leads to a direct proof of the Feigin-Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. As an application, we give an explicit solution of Vinberg's quantization problem by providing formulas for generators of maximal commutative subalgebras of $U(\mathfrak{g})$. We also calculate the eigenvalues of the Hamiltonians on the Bethe vectors in the Gaudin model associated with $\…
We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form. An almost Abelian Lie algebra has only one non-vanishing Lie bracket, which is given by a linear operator. If the latter is Jordanable then we call the almost Abelian Lie algebra Jordanable. The existence of canonical form allows to describe important structural properties of these Lie algebras in explicit terms. Lie subalgebras, ideals, automorphisms and derivations, as well as quadratic Casimir elements are given in explicitly in a suitable basis.
Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial construction of g, where root vectors correspond to vertices of a certain quiver $Gammahat$, and show that with respect to this basis the structure constants of the Lie bracket are given by paths in $Gammahat$. This construction is then related to the constructions of Ringel and Peng and Xiao.
In the first part of this paper, we discuss the classical W-algebra $\mathcal{W}(\mathfrak{g}, F)$ associated with a Lie superalgebra $\mathfrak{g}$ and the nilpotent element $F$ in an $\mathfrak{sl}_2$-triple. We find a generating set of $\mathcal{W}(\mathfrak{g}, F)$ and compute the Poisson brackets between them. In the second part, which is the main part of the paper, we discuss supersymmetric classical W-algebras. We introduce two different constructions of a supersymmetric classical W-algebra $\mathcal{W}(\mathfrak{g}, f)$ associated with a Lie superalgebra $\mathfrak{g}$ and an odd nilpo…
The crystals for a finite-dimensional complex reductive Lie algebra $\mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{\mathfrak{g}}$, constructed using the Dynkin diagram of $\mathfrak{g}$, and its combinatorial action on any $\mathfrak{g}$-crystal via Schützenberger involutions. We compare this action with that of the Berenstein-Kirillov group on Gelfand-Tsetlin patterns. Henriques and Kamnitzer define an action of $C_n=C_{\mathfrak{gl}_n}$ on $n$-tensor products of $\mathfrak{g}$-crystals, for …
We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations of all distinguished pre-Nichols algebras arXiv:1405.6681 belonging to a one-parameter family; we call these algebras \emph{large} quantum groups. We prove that each of these quantum algebras has a central Hopf subalgebra giving rise to a Poisson order in the sense of arXiv:math/0201042.. We describe explicitly the underlying Poisson algebraic groups and Po…
Mathematics Institute, Georg-August-University of Göttingen（哥廷根大学数学研究所）；Department of Mathematics, Aristotle University of Thessaloniki（萨洛尼卡亚里士多德大学数学系）；Institut für Mathematik, Julius-Maximilians-Universität Würzburg（维尔茨堡大学数学研究所）
