Universal Enveloping Algebras and Some Applications in Physics
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Commuting Ordinary Differential Operators and the Dixmier Test
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 101, 23 pages Commuting Ordinary Differential Operators and the Dixmier Test Emma PREVIATO y, Sonia L. RUEDA z and Maria-Angeles ZURRO x y Boston University, USA E-mail: [email protected] z Universidad Polit´ecnica de Madrid, Spain E-mail: [email protected] x Universidad Aut´onomade Madrid, Spain E-mail: [email protected] Received February 04, 2019, in final form December 23, 2019; Published online December 30, 2019 https://doi.org/10.3842/SIGMA.2019.101 Abstract. The Burchnall{Chaundy problem is classical in differential algebra, seeking to describe all commutative subalgebras of a ring of ordinary differential operators whose coefficients are functions in a given class. It received less attention when posed in the (first) Weyl algebra, namely for polynomial coefficients, while the classification of commutative subalgebras of the Weyl algebra is in itself an important open problem. Centralizers are maximal-commutative subalgebras, and we review the properties of a basis of the centralizer of an operator L in normal form, following the approach of K.R. Goodearl, with the ultimate goal of obtaining such bases by computational routines. Our first step is to establish the Dixmier test, based on a lemma by J. Dixmier and the choice of a suitable filtration, to give necessary conditions for an operator M to be in the centralizer of L. Whenever the centralizer equals the algebra generated by L and M, we call L, M a Burchnall{Chaundy (BC) pair. A construction of BC pairs is presented for operators of order 4 in the first Weyl algebra. -
Arxiv:1705.02958V2 [Math-Ph]
LMU-ASC 30/17 HOCHSCHILD COHOMOLOGY OF THE WEYL ALGEBRA AND VASILIEV’S EQUATIONS ALEXEY A. SHARAPOV AND EVGENY D. SKVORTSOV Abstract. We propose a simple injective resolution for the Hochschild com- plex of the Weyl algebra. By making use of this resolution, we derive ex- plicit expressions for nontrivial cocycles of the Weyl algebra with coefficients in twisted bimodules as well as for the smash products of the Weyl algebra and a finite group of linear symplectic transformations. A relationship with the higher-spin field theory is briefly discussed. 1. Introduction The polynomial Weyl algebra An is defined to be an associative, unital algebra over C on 2n generators yj subject to the relations yj yk − ykyj =2iπjk. Here π = (πjk) is a nondegenerate, anti-symmetric matrix over C. Bringing the ⊗n matrix π into the canonical form, one can see that An ≃ A1 . The Hochschild (co)homology of the Weyl algebra is usually computed employing a Koszul-type resolution, see e.g. [1], [2], [3]. More precisely, the Koszul complex of the Weyl algebra is defined as a finite subcomplex of the normalized bar-resolution, so that the restriction map induces an isomorphism in (co)homology [3]. This makes it relatively easy to calculate the dimensions of the cohomology spaces for various coefficients. Finding explicit expressions for nontrivial cocycles appears to be a much more difficult problem. For example, it had long been known that • ∗ 2n ∗ HH (An, A ) ≃ HH (An, A ) ≃ C . arXiv:1705.02958v2 [math-ph] 6 Sep 2017 n n (The only nonzero group is dual to the homology group HH2n(An, An) generated by 1 2 2n the cycle 1⊗y ∧y ∧···∧y .) An explicit formula for a nontrivial 2n-cocycle τ2n, however, remained unknown, until it was obtained in 2005 paper by Feigin, Felder and Shoikhet [4] as a consequence of the Tsygan formality for chains [5], [6]. -
1 the Basic Set-Up 2 Poisson Brackets
MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016–2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical Mechanics, Chapter 14. 1 The basic set-up I assume that you have already studied Gregory, Sections 14.1–14.4. The following is intended only as a succinct summary. We are considering a system whose equations of motion are written in Hamiltonian form. This means that: 1. The phase space of the system is parametrized by canonical coordinates q =(q1,...,qn) and p =(p1,...,pn). 2. We are given a Hamiltonian function H(q, p, t). 3. The dynamics of the system is given by Hamilton’s equations of motion ∂H q˙i = (1a) ∂pi ∂H p˙i = − (1b) ∂qi for i =1,...,n. In these notes we will consider some deeper aspects of Hamiltonian dynamics. 2 Poisson brackets Let us start by considering an arbitrary function f(q, p, t). Then its time evolution is given by n df ∂f ∂f ∂f = q˙ + p˙ + (2a) dt ∂q i ∂p i ∂t i=1 i i X n ∂f ∂H ∂f ∂H ∂f = − + (2b) ∂q ∂p ∂p ∂q ∂t i=1 i i i i X 1 where the first equality used the definition of total time derivative together with the chain rule, and the second equality used Hamilton’s equations of motion. The formula (2b) suggests that we make a more general definition. Let f(q, p, t) and g(q, p, t) be any two functions; we then define their Poisson bracket {f,g} to be n def ∂f ∂g ∂f ∂g {f,g} = − . -
Poisson Structures and Integrability
Poisson Structures and Integrability Peter J. Olver University of Minnesota http://www.math.umn.edu/ olver ∼ Hamiltonian Systems M — phase space; dim M = 2n Local coordinates: z = (p, q) = (p1, . , pn, q1, . , qn) Canonical Hamiltonian system: dz O I = J H J = − dt ∇ ! I O " Equivalently: dpi ∂H dqi ∂H = = dt − ∂qi dt ∂pi Lagrange Bracket (1808): n ∂pi ∂qi ∂qi ∂pi [ u , v ] = ∂u ∂v − ∂u ∂v i#= 1 (Canonical) Poisson Bracket (1809): n ∂u ∂v ∂u ∂v u , v = { } ∂pi ∂qi − ∂qi ∂pi i#= 1 Given functions u , . , u , the (2n) (2n) matrices with 1 2n × respective entries [ u , u ] u , u i, j = 1, . , 2n i j { i j } are mutually inverse. Canonical Poisson Bracket n ∂F ∂H ∂F ∂H F, H = F T J H = { } ∇ ∇ ∂pi ∂qi − ∂qi ∂pi i#= 1 = Poisson (1809) ⇒ Hamiltonian flow: dz = z, H = J H dt { } ∇ = Hamilton (1834) ⇒ First integral: dF F, H = 0 = 0 F (z(t)) = const. { } ⇐⇒ dt ⇐⇒ Poisson Brackets , : C∞(M, R) C∞(M, R) C∞(M, R) { · · } × −→ Bilinear: a F + b G, H = a F, H + b G, H { } { } { } F, a G + b H = a F, G + b F, H { } { } { } Skew Symmetric: F, H = H, F { } − { } Jacobi Identity: F, G, H + H, F, G + G, H, F = 0 { { } } { { } } { { } } Derivation: F, G H = F, G H + G F, H { } { } { } F, G, H C∞(M, R), a, b R. ∈ ∈ In coordinates z = (z1, . , zm), F, H = F T J(z) H { } ∇ ∇ where J(z)T = J(z) is a skew symmetric matrix. − The Jacobi identity imposes a system of quadratically nonlinear partial differential equations on its entries: ∂J jk ∂J ki ∂J ij J il + J jl + J kl = 0 ! ∂zl ∂zl ∂zl " #l Given a Poisson structure, the Hamiltonian flow corresponding to H C∞(M, R) is the system of ordinary differential equati∈ons dz = z, H = J(z) H dt { } ∇ Lie’s Theory of Function Groups Used for integration of partial differential equations: F , F = G (F , . -
Lecture 3 1.1. a Lie Algebra Is a Vector Space Along with a Map [.,.] : 多 多 多 Such That, [Αa+Βb,C] = Α[A,C]+Β[B,C] B
Lecture 3 1. LIE ALGEBRAS 1.1. A Lie algebra is a vector space along with a map [:;:] : L ×L ! L such that, [aa + bb;c] = a[a;c] + b[b;c] bi − linear [a;b] = −[b;a] Anti − symmetry [[a;b];c] + [[b;c];a][[c;a];b] = 0; Jacobi identity We will only think of real vector spaces. Even when we talk of matrices with complex numbers as entries, we will assume that only linear combina- tions with real combinations are taken. 1.1.1. A homomorphism is a linear map among Lie algebras that preserves the commutation relations. 1.1.2. An isomorphism is a homomorphism that is invertible; that is, there is a one-one correspondence of basis vectors that preserves the commuta- tion relations. 1.1.3. An homomorphism to a Lie algebra of matrices is called a represe- tation. A representation is faithful if it is an isomorphism. 1.2. Examples. (1) The basic example is the cross-product in three dimensional Eu- clidean space. Recall that i j k a × b = a1 a2 a3 b1 b2 b3 The bilinearity and anti-symmetry are obvious; the Jacobi identity can be verified through tedious calculations. Or you can use the fact that any cross product is determined by the cross-product of the basis vectors through linearity; and verify the Jacobi identity on the basis vectors using the cross products i × j = k; j × k = i; k×i= j Under many different names, this Lie algebra appears everywhere in physics. It is the single most important example of a Lie algebra. -
Lecture 21: Symmetric Products and Algebras
LECTURE 21: SYMMETRIC PRODUCTS AND ALGEBRAS Symmetric Products The last few lectures have focused on alternating multilinear functions. This one will focus on symmetric multilinear functions. Recall that a multilinear function f : U ×m ! V is symmetric if f(~v1; : : : ;~vi; : : : ;~vj; : : : ;~vm) = f(~v1; : : : ;~vj; : : : ;~vi; : : : ;~vm) for any i and j, and for any vectors ~vk. Just as with the exterior product, we can get the universal object associated to symmetric multilinear functions by forming various quotients of the tensor powers. Definition 1. The mth symmetric power of V , denoted Sm(V ), is the quotient of V ⊗m by the subspace generated by ~v1 ⊗ · · · ⊗ ~vi ⊗ · · · ⊗ ~vj ⊗ · · · ⊗ ~vm − ~v1 ⊗ · · · ⊗ ~vj ⊗ · · · ⊗ ~vi ⊗ · · · ⊗ ~vm where i and j and the vectors ~vk are arbitrary. Let Sym(U ×m;V ) denote the set of symmetric multilinear functions U ×m to V . The following is immediate from our construction. Lemma 1. We have an natural bijection Sym(U ×m;V ) =∼ L(Sm(U);V ): n We will denote the image of ~v1 ⊗: : :~vm in S (V ) by ~v1 ·····~vm, the usual notation for multiplication to remind us that the terms here can be rearranged. Unlike with the exterior product, it is easy to determine a basis for the symmetric powers. Theorem 1. Let f~v1; : : : ;~vmg be a basis for V . Then _ f~vi1 · :::~vin ji1 ≤ · · · ≤ ing is a basis for Sn(V ). Before we prove this, we note a key difference between this and the exterior product. For the exterior product, we have strict inequalities. For the symmetric product, we have non-strict inequalities. -
The Weyl Algebras
THE WEYL ALGEBRAS David Cock Supervisor: Dr. Daniel Chan School of Mathematics, The University of New South Wales. November 2004 Submitted in partial fulfillment of the requirements of the degree of Bachelor of Science with Honours Contents Chapter 1 Introduction 1 Chapter 2 Basic Results 3 Chapter 3 Gradings and Filtrations 16 Chapter 4 Gelfand-Kirillov Dimension 21 Chapter 5 Automorphisms of A1 29 References 53 i Chapter 1 Introduction An important result in single-variable calculus is the so-called product rule. That is, for two polynomials (or more generally, functions) f(x), g(x): R → R: δ δ δ (fg) = ( f)g + f( g) δx δx δx It turns out that this formula, which is firmly rooted in calculus has very inter- esting algebraic properties. If k[x] denotes the ring of polynomials in one variable over a characteristic 0 field k, differentiation (in the variable x) can be considered as a map δ : k[x] → k[x]. It is relatively straighforward to verify that the map δ is in fact a k-linear vector space endomorphism of k[x]. Similarly, we can define another k-linear endomorphism X by left multiplication by x ie. X(f) = xf. Consider the expression (δ · X)f(x). Expanding this gives: (δ · X)f(x) = δ(xf(x)) applying the product rule gives (δ · X)f(x) = δ(x)f(x) + xδf(x) = f(x) + (X · δ)f(x) 1 noting the common factor of f(x) gives us the relation (this time in the ring of k-linear endomorphisms of k[x]): δ · X = X · δ + 1 where 1 is the identity map. -
QUANTUM WEYL ALGEBRAS and DEFORMATIONS of U(G)
Pacific Journal of Mathematics QUANTUM WEYL ALGEBRAS AND DEFORMATIONS OF U.g/ NAIHUAN JING AND JAMES ZHANG Volume 171 No. 2 December 1995 PACIFIC JOURNAL OF MATHEMATICS Vol. 171, No. 2, 1995 QUANTUM WEYL ALGEBRAS AND DEFORMATIONS OF U(G) NAIHUAN JING AND JAMES ZHANG We construct new deformations of the universal enveloping algebras from the quantum Weyl algebras for any R-matrix. Our new algebra (in the case of g = SI2) is a noncommuta- tive and noncocommutative bialgebra (i.e. quantum semi- group) with its localization being a Hopf algebra (i.e. quan- tum group). The ring structure and representation theory of our algebra are studied in the case of SI2. Introduction. Quantum groups are usually considered to be examples of ^-deformations of universal enveloping algebras of simple Lie algebras or their restricted dual algebras. The simplest example is the Drinfeld-Jimbo quantum group λ C/g(sl2), which is an associative algebra over C(q) generated by e,/, k,k~ subject to the following relations: ke = q2ek, kf = q~2fk, k-k-1 ef-fe =q-q-1 One sees that the element k does not stay in the same level as the elements e and / do, since k is actually an exponential of the original element h in the Cartan subalgebra. There do exist deformations to deform all generators into the same level, for example, Sklyanin algebras. However it is still uncertain if Sklyanin algebra is a bialgebra or not. Using the quantum Weyl algebras discussed in [WZ, GZ], we will con- struct new deformations for the enveloping algebras which have such a nice property. -
On Automorphisms of Weyl Algebra Bulletin De La S
BULLETIN DE LA S. M. F. L. MAKAR-LIMANOV On automorphisms of Weyl algebra Bulletin de la S. M. F., tome 112 (1984), p. 359-363 <http://www.numdam.org/item?id=BSMF_1984__112__359_0> © Bulletin de la S. M. F., 1984, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bull. Soc. math. France, 112, 1984, p. 359-363 ON AUTOMORPHISMS OF WEYL ALGEBRA BY L. MAKAR-LIMANOV (*) RESUME. — Le but de ccttc note est dc gcncraliscr un thcorcme de Jacques Dixmicr qui decrit le groupe des automorphismes de Falgebre de Weyl dans le cas de caracteristique positive et de donner une nouvelle demonstration dans le cas de caracteristique zero. ABSTRACT. — In this note I am going to generalize a theorem of J. Dixmier [1] describing the group of automorphisms of the Weyl algebra to the case of non-zero characteristic and give a new proof of it in the case of zero characteristic. Let A be the Weyl algebra with generators p and q ([p, q]sspq-'qpssi 1) over a field K. Let the characteristic of K be ^. LEMMA 1. — J/x 96 0 then p1 and q1 generate (over K) the center C of A. -
Left-Symmetric Algebras of Derivations of Free Algebras
LEFT-SYMMETRIC ALGEBRAS OF DERIVATIONS OF FREE ALGEBRAS Ualbai Umirbaev1 Abstract. A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of deriva- tions are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric al- gebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described. Mathematics Subject Classification (2010): Primary 17D25, 17A42, 14R15; Sec- ondary 17A36, 17A50. Key words: left-symmetric algebras, free algebras, derivations, Jacobian matrices. 1. Introduction If A is an arbitrary algebra over a field k, then the set DerkA of all k-linear derivations of A forms a Lie algebra. If A is a free algebra, then it is possible to define a multiplication · on DerkA such that it becomes a left-symmetric algebra and its commutator algebra becomes the Lie algebra DerkA of all derivations of A. The language of the left-symmetric algebras of derivations is more convenient to describe some combinatorial properties of derivations. Recall that an algebra B over k is called left-symmetric [4] if B satisfies the identity (1) (xy)z − x(yz)=(yx)z − y(xz). This means that the associator (x, y, z) := (xy)z −x(yz) is symmetric with respect to two left arguments, i.e., (x, y, z)=(y, x, z). -
WOMP 2001: LINEAR ALGEBRA Reference Roman, S. Advanced
WOMP 2001: LINEAR ALGEBRA DAN GROSSMAN Reference Roman, S. Advanced Linear Algebra, GTM #135. (Not very good.) 1. Vector spaces Let k be a field, e.g., R, Q, C, Fq, K(t),. Definition. A vector space over k is a set V with two operations + : V × V → V and · : k × V → V satisfying some familiar axioms. A subspace of V is a subset W ⊂ V for which • 0 ∈ W , • If w1, w2 ∈ W , a ∈ k, then aw1 + w2 ∈ W . The quotient of V by the subspace W ⊂ V is the vector space whose elements are subsets of the form (“affine translates”) def v + W = {v + w : w ∈ W } (for which v + W = v0 + W iff v − v0 ∈ W , also written v ≡ v0 mod W ), and whose operations +, · are those naturally induced from the operations on V . Exercise 1. Verify that our definition of the vector space V/W makes sense. Given a finite collection of elements (“vectors”) v1, . , vm ∈ V , their span is the subspace def hv1, . , vmi = {a1v1 + ··· amvm : a1, . , am ∈ k}. Exercise 2. Verify that this is a subspace. There may sometimes be redundancy in a spanning set; this is expressed by the notion of linear dependence. The collection v1, . , vm ∈ V is said to be linearly dependent if there is a linear combination a1v1 + ··· + amvm = 0, some ai 6= 0. This is equivalent to being able to express at least one of the vi as a linear combination of the others. Exercise 3. Verify this equivalence. Theorem. Let V be a vector space over a field k. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group).