APPENDIX a Tensors, Filtrations, and Gradings
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1. Introduction There Is a Deep Connection Between Algebras and the Categories of Their Modules
Pr´e-Publica¸c~oesdo Departamento de Matem´atica Universidade de Coimbra Preprint Number 09{15 SEMI-PERFECT CATEGORY-GRADED ALGEBRAS IVAN YUDIN Abstract: We introduce the notion of algebras graded over a small category and give a criterion for such algebras to be semi-perfect. AMS Subject Classification (2000): 18E15,16W50. 1. Introduction There is a deep connection between algebras and the categories of their modules. The interplay between their properties contributes to both the structure theory of algebras and the theory of abelian categories. It is often the case that the study of the category of modules over a par- ticular algebra can lead to the use of other abelian categories, which are non-equivalent to module categories over any algebra. One of such exam- ples is the category of modules over a C-graded algebra, where C is a small category. Such categories appear in the joint work of the author and A. P. Santana [18] on homological properties of Schur algebras. Algebras graded over a small category generalise the widely known group graded algebras (see [16, 12, 8, 15, 14]), the recently introduced groupoid graded algebras (see [10, 11, 9]), and Z-algebras, used in the theory of operads (see [17]). One of the important properties of some abelian categories, that consider- ably simplifies the study of their homological properties, is the existence of projective covers for finitely generated objects. Such categories were called semi-perfect in [7]. We give in this article the characterisation of C-graded algebras whose categories of modules are semi-perfect. -
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. -
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). -
Z3-Graded Geometric Algebra 1 Introduction
Adv. Studies Theor. Phys., Vol. 4, 2010, no. 8, 383 - 392 Z3-Graded Geometric Algebra Farid Makhsoos Department of Mathematics Faculty of Sciences Islamic Azad University of Varamin(Pishva) Varamin-Tehran-Iran Majid Bashour Department of Mathematics Faculty of Sciences Islamic Azad University of Varamin(Pishva) Varamin-Tehran-Iran [email protected] Abstract By considering the Z2 gradation structure, the aim of this paper is constructing Z3 gradation structure of multivectors in geometric alge- bra. Mathematics Subject Classification: 16E45, 15A66 Keywords: Graded Algebra, Geometric Algebra 1 Introduction The foundations of geometric algebra, or what today is more commonly known as Clifford algebra, were put forward already in 1844 by Grassmann. He introduced vectors, scalar products and extensive quantities such as exterior products. His ideas were far ahead of his time and formulated in an abstract and rather philosophical form which was hard to follow for contemporary math- ematicians. Because of this, his work was largely ignored until around 1876, when Clifford took up Grassmann’s ideas and formulated a natural algebra on vectors with combined interior and exterior products. He referred to this as an application of Grassmann’s geometric algebra. Due to unfortunate historic events, such as Clifford’s early death in 1879, his ideas did not reach the wider part of the mathematics community. Hamil- ton had independently invented the quaternion algebra which was a special 384 F. Makhsoos and M. Bashour case of Grassmann’s constructions, a fact Hamilton quickly realized himself. Gibbs reformulated, largely due to a misinterpretation, the quaternion alge- bra to a system for calculating with vectors in three dimensions with scalar and cross products. -
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. -
Universal Enveloping Algebras and Some Applications in Physics
Universal enveloping algebras and some applications in physics Xavier BEKAERT Institut des Hautes Etudes´ Scientifiques 35, route de Chartres 91440 – Bures-sur-Yvette (France) Octobre 2005 IHES/P/05/26 IHES/P/05/26 Universal enveloping algebras and some applications in physics Xavier Bekaert Institut des Hautes Etudes´ Scientifiques Le Bois-Marie, 35 route de Chartres 91440 Bures-sur-Yvette, France [email protected] Abstract These notes are intended to provide a self-contained and peda- gogical introduction to the universal enveloping algebras and some of their uses in mathematical physics. After reviewing their abstract definitions and properties, the focus is put on their relevance in Weyl calculus, in representation theory and their appearance as higher sym- metries of physical systems. Lecture given at the first Modave Summer School in Mathematical Physics (Belgium, June 2005). These lecture notes are written by a layman in abstract algebra and are aimed for other aliens to this vast and dry planet, therefore many basic definitions are reviewed. Indeed, physicists may be unfamiliar with the daily- life terminology of mathematicians and translation rules might prove to be useful in order to have access to the mathematical literature. Each definition is particularized to the finite-dimensional case to gain some intuition and make contact between the abstract definitions and familiar objects. The lecture notes are divided into four sections. In the first section, several examples of associative algebras that will be used throughout the text are provided. Associative and Lie algebras are also compared in order to motivate the introduction of enveloping algebras. The Baker-Campbell- Haussdorff formula is presented since it is used in the second section where the definitions and main elementary results on universal enveloping algebras (such as the Poincar´e-Birkhoff-Witt) are reviewed in details. -
Dimension Formula for Graded Lie Algebras and Its Applications
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 11, Pages 4281–4336 S 0002-9947(99)02239-4 Article electronically published on June 29, 1999 DIMENSION FORMULA FOR GRADED LIE ALGEBRAS AND ITS APPLICATIONS SEOK-JIN KANG AND MYUNG-HWAN KIM Abstract. In this paper, we investigate the structure of infinite dimensional Lie algebras L = α Γ Lα graded by a countable abelian semigroup Γ sat- isfying a certain finiteness∈ condition. The Euler-Poincar´e principle yields the denominator identitiesL for the Γ-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces Lα (α Γ). Our dimen- sion formula enables us to study the structure of the Γ-graded2 Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also dis- cuss the relation of graded Lie algebras and the product identities for formal power series. Introduction In [K1] and [Mo1], Kac and Moody independently introduced a new class of Lie algebras, called Kac-Moody algebras, as a generalization of finite dimensional simple Lie algebras over C. These algebras are ususally infinite dimensional, and are defined by the generators and relations associated with generalized Cartan matrices, which is similar to Serre’s presentation of complex semisimple Lie algebras. In [M], by generalizing Weyl’s denominator identity to the affine root system, Macdonald obtained a new family of combinatorial identities, including Jacobi’s triple product identity as the simplest case. -
Arxiv:Math/9810152V2 [Math.RA] 12 Feb 1999 Hoe 0.1
GORENSTEINNESS OF INVARIANT SUBRINGS OF QUANTUM ALGEBRAS Naihuan Jing and James J. Zhang Abstract. We prove Auslander-Gorenstein and GKdim-Macaulay properties for certain invariant subrings of some quantum algebras, the Weyl algebras, and the universal enveloping algebras of finite dimensional Lie algebras. 0. Introduction Given a noncommutative algebra it is generally difficult to determine its homological properties such as global dimension and injective dimension. In this paper we use the noncommutative version of Watanabe theorem proved in [JoZ, 3.3] to give some simple sufficient conditions for certain classes of invariant rings having some good homological properties. Let k be a base field. Vector spaces, algebras, etc. are over k. Suppose G is a finite group of automorphisms of an algebra A. Then the invariant subring is defined to be AG = {x ∈ A | g(x)= x for all g ∈ G}. Let A be a filtered ring with a filtration {Fi | i ≥ 0} such that F0 = k. The associated graded ring is defined to be Gr A = n Fn/Fn−1. A filtered (or graded) automorphism of A (or Gr A) is an automorphism preservingL the filtration (or the grading). The following is a noncommutative version of [Ben, 4.6.2]. Theorem 0.1. Suppose A is a filtered ring such that the associated graded ring Gr A is isomorphic to a skew polynomial ring kpij [x1, · · · , xn], where pij 6= pkl for all (i, j) 6=(k, l). Let G be a finite group of filtered automorphisms of A with |G| 6=0 in k. If det g| n =1 for all g ∈ G, then (⊕i=1kxi) AG is Auslander-Gorenstein and GKdim-Macaulay. -
Clifford Algebra Analogue of the Hopf–Koszul–Samelson Theorem
Advances in Mathematics AI1608 advances in mathematics 125, 275350 (1997) article no. AI971608 Clifford Algebra Analogue of the HopfKoszulSamelson Theorem, the \-Decomposition C(g)=End V\ C(P), and the g-Module Structure of Ãg Bertram Kostant* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received October 18, 1996 1. INTRODUCTION 1.1. Let g be a complex semisimple Lie algebra and let h/b be, respectively, a Cartan subalgebra and a Borel subalgebra of g. Let n=dim g and l=dim h so that n=l+2r where r=dim gÂb. Let \ be one- half the sum of the roots of h in b. The irreducible representation ?\ : g Ä End V\ of highest weight \ is dis- tinguished among all finite-dimensional irreducible representations of g for r a number of reasons. One has dim V\=2 and, via the BorelWeil theorem, Proj V\ is the ambient variety for the minimal projective embed- ding of the flag manifold associated to g. In addition, like the adjoint representation, ?\ admits a uniform construction for all g. Indeed let SO(g) be defined with respect to the Killing form Bg on g. Then if Spin ad: g Ä End S is the composite of the adjoint representation ad: g Ä Lie SO(g) with the spin representation Spin: Lie SO(g) Ä End S, it is shown in [9] that Spin ad is primary of type ?\ . Given the well-known relation between SS and exterior algebras, one immediately deduces that the g-module structure of the exterior algebra Ãg, with respect to the derivation exten- sion, %, of the adjoint representation, is given by l Ãg=2 V\V\.(a) The equation (a) is only skimming the surface of a much richer structure. -
Tensor, Exterior and Symmetric Algebras
Tensor, Exterior and Symmetric Algebras Daniel Murfet May 16, 2006 Throughout this note R is a commutative ring, all modules are left R-modules. If we say a ring is noncommutative, we mean it is not necessarily commutative. Unless otherwise specified, all rings are noncommutative (except for R). If A is a ring then the center of A is the set of all x ∈ A with xy = yx for all y ∈ A. Contents 1 Definitions 1 2 The Tensor Algebra 5 3 The Exterior Algebra 6 3.1 Dimension of the Exterior Powers ............................ 11 3.2 Bilinear Forms ...................................... 14 3.3 Other Properties ..................................... 18 3.3.1 The determinant formula ............................ 18 4 The Symmetric Algebra 19 1 Definitions Definition 1. A R-algebra is a ring morphism φ : R −→ A where A is a ring and the image of φ is contained in the center of A. This is equivalent to A being an R-module and a ring, with r · (ab) = (r · a)b = a(r · b), via the identification of r · 1 and φ(r). A morphism of R-algebras is a ring morphism making the appropriate diagram commute, or equivalently a ring morphism which is also an R-module morphism. In this section RnAlg will denote the category of these R-algebras. We use RAlg to denote the category of commutative R-algebras. A graded ring is a ring A together with a set of subgroups Ad, d ≥ 0 such that A = ⊕d≥0Ad as an abelian group, and st ∈ Ad+e for all s ∈ Ad, t ∈ Ae. -
Determinants of Complexes
APPENDIX A Determinants of Complexes Complexes Let k be a field. A complex of k-vector spaces is a graded vector space W· = EBieZ Wi together with a family of linear operators di+l Odi = o. The spaces Wi are called the terms of W·. Throughout this Appendix we shall assume that all but finitely many tenns of a complex are zero. Cohomology spaces of a complex w· are defined as Hi (W·) = Ker(di)/Im(di_l). A complex w· is called exact if all Hi (W·) are equal to zero. Example 1. Any linear operator between two vector spaces, say, w- 1 and Wo, can be regarded as a complex with only two non-zero tenns: Such a complex is exact if and only if d_1 is invertible. For any graded vector space w· and any m E Z we shall denote by W·[m] the same vector space but with the grading shifted by m: The same notation will be used for complexes. Let v· and w· be two complexes of vector spaces. A morphism of complexes J : V· -+ w· is a collection of linear operators /; : Vi -+ Wi which commute with the differentials in v· and w·. The cone of a morphism J is a new complex Cone(f) with tenns Cone(fi = Wi 6) Vi+l and the differential given by d(w, v) = (dw(w) + (-li+1J(v), dv(v»), WE Wi, V E Vi+l. (I) Any morphism of complexes J : V· -+ w· induces a morphism of cohomology spaces Hi (f) : Hi(V.) -+ Hi(W·). -
Arxiv:Math/0303372V1
KOSTANT THEOREM FOR SPECIAL FILTERED ALGEBRAS VYACHESLAV FUTORNY AND SERGE OVSIENKO Abstract. A famous result of Kostant states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. We prove an analogue of this result for the class of special filtered algebras and apply it to show the freeness over its center of the restricted Yangian and the universal enveloping algebra of the restricted current algebra, associated with the general linear Lie algebra. Mathematics Subject Classification 13A02, 16W70, 17B37, 81R10 1. Introduction A well-known theorem of Kostant [K] says that for a complex semisimple Lie algebra g the universal enveloping algebra U(g) is a free module over its center. For the Yangian Y(gln) of the general linear Lie algebra gln the freeness over the center was shown in [MNO]. Another example of such situation gives the Gelfand-Tsetlin subalgebra Γ of U(gln) (see [DFO], [O1]). It was shown in [O2] that U(gln) is a free left (right) Γ−module. The knowledge of the freeness of a given algebra over its certain subalgebra often allows to establish other important results, e.g. to find the annihilators of Verma modules [Di], to prove the finiteness of the number of liftings to an irreducible module over U(gln) from a given character of the Gelfand-Tsetlin subalgebra ([O2]), etc. In [O2] was introduced a technique which generalizes Kostant methods ([K], see also [G1]) and which allows to study the universal enveloping algebras of Lie algebras as modules over its certain commutative subalgebras.