AAn Invitation to Noncommutative Algebra
Chelsea Walton
Abstract
This is a brief introduction to the world of Noncommutative Algebraaimed at advanced undergraduate and beginning graduate students.
The purpose of this note is to invite you, the reader, into the world of Noncommu-tative Algebra. What is it? In short, it is the study of algebraic structures that havea noncommutative multiplication. One’s first encounter with these structures occurstypically with matrices. Indeed, given two n -by- n matrices X and Y with n >
1, weget that XY (cid:54) = Y X in general. But this simple observation motivates a deeper reasonwhy Noncommutative Algebra is ubiquitous...Let’s consider two basic transformations of images in real 2-space: Rotation by90 degrees clockwise and Reflection about the vertical axis. As we see in the figuresbelow, the order in which these transformations are performed matters . Fig. 1: The composition of rotation and reflection transformations is noncommutative.Chelsea WaltonThe University of Illinois at Urbana-Champaign, Department of Mathematics, 273 Altgeld Hall,1409 W. Green Street, Urbana, IL 61801, e-mail: [email protected] a r X i v : . [ m a t h . HO ] F e b Chelsea Walton
Since these transformations are linear (i.e., in R , lines are sent to lines), they canbe encoded by 2-by-2 matrices with entries in R [2, Section 3.C]. Namely, • ◦ CW Rotation corresponds to (cid:0) − (cid:1) , which sends vector ( v v ) to (cid:0) v − v (cid:1) ; • Reflection about the y -axis is encoded by (cid:0) − (cid:1) , which sends ( v v ) to (cid:0) − v v (cid:1) .The composition of linear transformations is then encoded by matrix multiplication.So, the first row in Figure 1 is corresponds to (cid:0) − (cid:1)(cid:0) − (cid:1) = (cid:0) − − (cid:1) , which sends ( v v ) to (cid:0) − v − v (cid:1) . Yet the second row is given by (cid:0) − (cid:1)(cid:0) − (cid:1) = (cid:0) (cid:1) , sending ( v v ) to ( v v ) . Therefore, the outcome of Figure 1 is a result of the fact that (cid:0) − − (cid:1) (cid:54) = (cid:0) (cid:1) .One can cook up other, say higher dimensional, examples of the varying out-comes of composing linear transformations by exploiting the noncommutativity ofmatrix multiplication. This is all part of the general phenomenon that functions donot commute under composition typically. (Think of the myriad of outcomes ofcomposing functions from everyday life– for instance, washing and drying clothes!)Now let’s turn our attention to special functions that we first encounter as chil-dren: Symmetries . To make this concept more concrete mathematically, considerthe informal definition and notation below.
Definition 1.
Take any object X . Then, a symmetry of X is an invertible, structure/property-preserving transformation from X to itself. The collection of such transfor-mations is denoted by Sym( X ).Historically, the examination of symmetries in mathematics and physics servedas one of the inspirations for the defining a group as an abstract algebraic structure(see, e.g., [43, Section 1(c)]). Namely, Sym( X ) is a group with the identity element e being the “do nothing” transformation, with composition as the associative binaryoperation, and Sym( X ) is equipped with inverse elements by definition.Continuing the example above: Take X = R and Sym( R ) to be the collection of R -linear transformations from R to R (so the origin is fixed). We get that Sym( R )is the general linear group GL( R ), often written as GL ( R ) denoting the group ofall invertible 2-by-2 matrices with real entries. Further, this group is nonabelian ;thus, composition of R -linear symmetries of R is noncommutative.Another concept that is inherently noncommutative is that of a representation .We will see later in Section 3 that this is best motivated by elementary problem offinding matrix solutions to equations (which, in turn, can have physical implica-tions). But for now let’s think about the problem below. Question 1.
Which matrices M ∈ Mat ( R ) satisfy the equation x = M = (cid:0) a bc d (cid:1) andsolve for entries a , b , c , d that satisfy (cid:32) a bc d (cid:33) (cid:32) a bc d (cid:33) = (cid:32) (cid:33) . n Invitation to Noncommutative Algebra 3 Not only is this boring, and it can be very tedious to find solutions to more generalproblems (e.g., taking instead M ∈ Mat n ( R ) for n > T defined by the equation x =
1, and link T to Sym( R ) via a structure preserving map φ . Then, a solution toQuestion 1 is produced in terms of an image of φ .For example, we could take T to be the group Z as its presentation is given by (cid:104) x | x = e (cid:105) . An example of a structure-preserving map φ is given by φ : Z → Sym ( R ) , e (cid:55)→ { Do Nothing } , x (cid:55)→ { Reflection about y -axis } .Indeed, φ ( gg (cid:48) ) = φ ( g ) ◦ φ ( g (cid:48) ) for all g , g (cid:48) ∈ Z . For instance, φ ( x ) ◦ φ ( x ) = { Ref. about y -axis }◦{ Ref. about y -axis } = { Nothing } = φ ( e ) = φ ( x ) . Since φ ( e ) and φ ( x ) correspond respectively to (cid:0) (cid:1) and (cid:0) − (cid:1) , these matricesare solutions to Question 1. Further, other reflections of R produce additional so-lutions to Question 1. (Think about if all solutions to Question 1 can be constructedin this manner.) xy xy xy Ref. about y = 0 (cid:33) (cid:0) − (cid:1) Rotation 180 ◦ CW (cid:33) (cid:0) − − (cid:1) Ref. about y = x (cid:33) (cid:0) (cid:1) Fig. 2: Reflections of R and the corresponding solution to Question 1. Continuing this example, instead of using the abstract group Z we could haveused the group algebra T = RZ , as it encodes the same information needed to ad-dress Question 1. We will chat more about abstract algebraic structures in Section 2(see Figure 5); in any case, their representations are defined informally below. Definition 2.
Given an abstract algebraic structure T , we say that a representation of T is an object X equipped with a structure/ property-preserving map T → Sym ( X ) .An example of a representation of a group G is a vector space V equipped witha group homomorphism G → GL ( V ) , where GL ( V ) is the group of invertible lineartransformations from V to itself (e.g., GL ( R ) = GL ( R ) as discussed above). Justas a representation of G is identified as a G-module , representations of rings and ofalgebras coincide with modules over such structures (see Figure 12 below). See also[50, Chapters 1 and 3] for further reading and examples.Now
Representation Theory is essentially a noncommutative area due to the fol-lowing key fact. Take A to a be commutative algebra over a field k with a representa-tion V of A , that is, a k -vector space V equipped with algebra map φ : A → GL ( V ) . If ( V , φ ) is irreducible [Definition 14], then dim k V = Chelsea Walton
Moreover, Representation Theory is a vital subject because the problem of find-ing matrix solutions to equations is quite natural. Since this boils down down tostudying representations of algebras that are generally noncommutative, the ubiq-uity of Noncommutative Algebra is conceivable. (Equations that correspond to rep-resentations of groups, like in Question 1, are special.)To introduce the final notion in Noncommutative Algebra that we will highlightin this paper, observe symmetries and representations both occur under an action ofa gadget T on an object X , but the difference is that symmetries form the gadget T (what is acting on an object), whereas representations are considered to be the object X (something being acted upon). What happens to these notions if we consider deformations of T and X ? Consider the following informal terminology. Definition 3. A deformation of an object X is an object X def that has many of thesame characteristics of X , possibly with the exception of a few key features. Inparticular, a deformation of an algebraic structure T is an algebraic structure T def ofthe same type that shares a (less complex) underlying structure of T .For example, a deformation of a ring R could be another ring R def that equals R asabelian groups, possibly with a different multiplication than that of R (see Figure 5).Now if we deform an object X , is there a gadget T def that acts on X def naturally?On the other hand, if we deform the gadget T , is there a natural deformation X def of X that comes equipped with an action of T def ? These are obvious questions, yet theanswers are difficult to visualize. This is because, visually, symmetries of an object X are destroyed when X is altered, even slightly; see Figure 3 below. X : Equilateral triangle Isosceles triangle Scalene triangleSym( X ): Dihedral group of order 6 Cyclic group of order 2 Trivial group of order 1Fig. 3: Triangles and their respective symmetry groups So we need to think beyond what can be visualized and consider a larger math-ematical framework that handles symmetries under deformation. To do so, it is es-sential to think beyond group actions, because, many classes of groups, includingfinite groups, do not admit deformations. However, group algebras or function alge-bras on groups do admit deformations, so we include these gadgets in the improvedframework to study symmetries. We will see later in Section 4 that when symme-tries are recast in the setting where they could be preserved under deformation, otherinteresting and more general algebraic gadgets like bialgebras and Hopf algebras arise in the process. This is crucial in Noncommutative Algebra as some of the mostimportant rings, especially those arising in physics, are noncommutative deforma-tions of commutative rings; the symmetries of such deformations deserve attention. n Invitation to Noncommutative Algebra 5
Symmetries , Representations , and
Deformations will play a key rolethroughout this article, just as they do in Noncommutative Algebra.
The remainder of the paper is two-fold: first, we will review three historical snap-shots of how Noncommutative Algebra played a prominent role in mathematicsand physics. We will discuss William Rowen Hamilton’s Quaternions in Section 2and the birth of Quantum Mechanics in Section 3. We will also briefly discuss theemergence of Quantum Groups in Section 4, together with the concept of QuantumSymmetry. In Section 5 we present a couple of research avenues for further investi-gation. All of the material here is by no means exhaustive, and many references willbe provided throughout.
Can numbers be noncommutative? The best answer is, as always, “Sure, why not?”
Fig. 4: Numbers that weall know and love... butwe should love more!
In this section we will explore a number system that gen-eralizes both the systems of real numbers R and complexnumbers C . The key feature of this new collection of num-bers –the quaternions H – is that they have a noncommuta-tive multiplication! This feature caused a bit of ruckus forWilliam Rowan Hamilton (1805-1865) after his discoveryof the quaternions in the mid-19th century. “Quaternions came from Hamilton after really good work hadbeen done; and, though beautifully ingenious, have been an un-mixed evil to those who have touched them in any way...”– Lord Kelvin, 1892 Now what do we mean by a number system ? Looselyspeaking, it is a set of quantities used to measure or count(a collection of) objects, which is equipped with an algebraic structure [Figure 5].Since we should be able to add, subtract, multiply and divide numbers, we con-sider the following terminology.
Definition 4.
Fix n ∈ Z ≥ . An n-dimensional division algebra D over R consists ofthe set of n -tuples of real numbers a : = ( a , a , . . . , a n ) , a i ∈ R , with : = ( , , . . . , ) and a unique element designated as so that • we can add and subtract two n -tuples a and b component-wise to form a + b and a − b in D , respectively; • we can multiply a by a scalar λ ∈ R component-wise to form λ ∗ a in D ; • there is a rule for multiplying a and b to form a · b in D (this is not necessarilydone component-wise, nor does it need to be commutative); and • there is a rule for dividing a by b (cid:54) = to form a ÷ b in D ; Chelsea WaltonFig. 5: Some algebraic structures. Straight arrows denote structures increasing in complexity.Dashed arrows denote structures merging compatibly to form another structure. in such a way that(i) ( D , + , − , ) is an abelian group ,(ii) ( D , + , − , , ∗ ) is an R -vector space ,(iii) ( D , · , ) is an associative unital ring , and(iv) ( D , + , − , ∗ , · , , ) is an associative R - algebra all in a compatible fashion (e.g. · distributes over +, etc.).As one can imagine, there are not many of these gadgets floating around as theyhave a lot of structure. A 1 -dimensional division algebra D over R must be the field R itself. Moreover, a 2 -dimensional division algebra D over R is isomorphic to thefield of complex numbers C , where the pair ( a , a ) is identified with the element a + a i for i = −
1. The algebraic structure for the pairs then follows accordingly,
Fig. 6: The real line, and thecomplex plane visualized as R . e.g., the multiplication of C is given by ( a , a ) · ( b , b ) = ( a b − a b , a b + a b ) . Note that the 1- and 2- dimensional real division al-gebras, R and C , are commutative rings, and thesecan be viewed geometrically as in Figure 6.A natural question is then the following. Question 2.
What are the n -dimensional real divi-sion algebras for n ≥ n = n Invitation to Noncommutative Algebra 7Fig. 7: A failed attempt at a3-dimensional number system. His initial ideas were to use two imaginaryaxes i and j so that the 3-tuples ( a , a , a ) of a 3-dimensional number system correspondto a + a i + a j . However, he could not cookup rules that i and j should obey to make thiscollection of triples a valid division algebra[30, 55, 67]. According to some mathemati-cians, this obsession was quite ‘Mad’ [4, 60].Finally, on October 16th 1843, on a walkwith his wife in Dublin, Hamilton had a mo-ment of Eureka! In his words to his son Archibald, “An electric circuit seemed to close; and a spark flashed forth, the herald, as I foresawimmediately , of many long years to come of definitely directed thought and work [...] Norcould I resist the impulse, unphilosopical as it may have been, to cut with a knife on thestone of Broughham Bridge, as we passed it, the fundamental formula with the symbols i , j , k ; namely i = j = k = i jk = − , which contains the solution of the problem ...”– W. R. Hamilton, August 5th, 1865Fig. 8: Plaque on BroughamBridge in Dublin, recognizingHamilton’s invention Hamilton had discovered that day a number sys-tem generalizing both R and C , consisting of of real numbers, not constructed from tripletsas he had imagined for so long [30]. Definition 5.
The quaternions is a 4-dimensionalreal division algebra, denoted by H , comprised of4-tuples of real numbers a : = ( a , a , a , a ) , whichare identified as elements of the form a + a i + a j + a k , for a i ∈ R , where addition, subtraction and scalar multiplica-tion are performed component-wise, and multipli-cation and division are governed by the rule i = j = k = i jk = − . Observe that jk = i , whereas k j = − i . Therefore, H is a noncommutative ring! Chelsea Walton
In any case, notice that the multiplication rule of H is a bit complicated: ( a , a , a , a ) · ( b , b , b , b ) = a b − a b − a b − a b , a b + a b + a b − a b , a b − a b + a b + a b , a b + a b − a b + a b (1)... and let’s not commit this rule to memory. To circumvent this issue Hamilton gavethe quaternions a geometric realization that encodes their multiplication. Namely,for a : = a + a i + a j + a k ∈ H , let a be the “scalar” component of a , and (cid:42) a : = a i + a j + a k be the “vector” component of a .Then, the vector components are visualized as points/ vectors in R , whereas thescalar component is realized as an element of time . See, for instance, the footnoteon page 60 and other parts of the preface of [29] for Hamilton’s original thoughts onthe connection between the quaternions and the laws of space and time. (Yes, yes,this was all very controversial back then!)Hamilton then devised two vector operations, now known as the dot product ( • )and cross product ( × ) to make the multiplication rule of H more compact: a · b = (cid:104) a b − (cid:42) a • (cid:42) b (cid:105) + (cid:104) a (cid:42) b + b (cid:42) a + (cid:42) a × (cid:42) b (cid:105) , ∀ a , b ∈ H . (2)Not only is formula (2) easier to retain than (1), the (commutative) dot productand (noncommutative) cross product have appeared in various parts of mathematics Fig. 9: A successful attempt at a4-dimensional number system. and physics throughout the years, including ourmulti-variable calculus courses.Geometrically, the operations in H capturesymmetries of R [Definition 1]: Addition/ sub-straction, scalar multiplication, and multiplica-tion/ division correspond respectively to trans-lation, dilation, and rotation of vectors of R ;see, e.g., [29, page 272] and [45] for a discus-sion of rotation. To see rotations in action, firstnote that the length of a quaternion a is given by | a | : = (cid:113) a + a + a + a . Next, fix an axis of rotation (cid:42) n : = n i + n j + n k with | (cid:42) n | =
1, a quaternion of unit-length. Then, rotating a vector (cid:42) q about the axis (cid:42) n clockwise by θ radians (whenviewed from the origin) corresponds to conjugating (cid:42) q by the quaternion e θ (cid:42) n . It’shelpful to use here an extension of Euler’s formula, e θ (cid:42) n = cos ( θ ) + sin ( θ ) (cid:42) n , tounderstand the quaternion e θ (cid:42) n . An example is given in Figure 10 below. n Invitation to Noncommutative Algebra 9Fig. 10: Rotating vector k about axis j by π radians (cid:33) Conjugating k by quaternion e π j . Moreover, rotations of R can be encoded as a representation [Definition 2] ofthe multiplicative subgroup U ( H ) consisting of unit-length quaternions. Indeed, wehave a group homomorphism U ( H ) −→ GL ( R ) = GL ( R ) , given by a + a i + a j + a k with | a | = (cid:55)→ a + a − a − a a a − a a a a + a a a a + a a a − a + a − a a a − a a a a − a a a a + a a a − a − a + a . This geometric realization of H has many modern applications– we refer to thetext [44] for a nice self-contained discussion of applications to computer-aided de-sign, aerospace engineering, and other fields.Returning to Question 2, its answer is now given below. Theorem 1. [26], [62], [46, Theorem 13.12], [8], [42], [71]
The answer to Ques-tion 2 is Yes if and only if n = , , , . Such division algebras D are unique up toisomorphism in their dimension with isomorphism class represented by • the real numbers R for n = , • the complex numbers C for n = , • quaternions H for n = , • octonions O for n = .Here, D is commutative only when n = , , and is associative only when n = , , . So, Hamilton discovered the last associative finite-dimensional real division al-gebra, but the price that he had to pay (at least mathematically) was the loss ofcommutativity. Perhaps this was not too high of a price– we are certainly willing tolose ordering when choosing to work with C instead of R . If we are also willing topart with associativity, then the octonions O is a perfectly suitable number system;see [3] for more details. And, of course, there are further generalizations of numbersystems– see [19, 49, 70] to start, and go wild!We return to the quaternions later in Section 5.1 for a discussion of potentialresearch directions. Another period that sparked an interest in Noncommutative Algebra was the birthof Quantum Mechanics in the 1920s. Three of the key figures during this time wereMax Born (1882-1970), Werner Heisenberg (1901-1976), and Paul Dirac (1902-1984), who were all curious about the behavior of subatomic particles [7, 32, 20].
Fig. 11: “More than anything, this photograph was really the result of a series of little accidents.”– Billy Huynh, photographer ...
So is good mathematics!
Along with their colleagues, Born, Heisenberg, and Dirac believed that impor-tant aspects of subatomic behavior are those that could be observed (or measured).However, the tools of classical mechanics available at the time (with observables corresponding to real-valued functions) were not suitable in capturing this behav-ior properly. A new type of mechanics was needed, leading to the development ofquantum mechanics where observables are realized as linear operators. For a greataccount of how this transition took place (some of which we will recall briefly be-low), see Part II of the van der Waerden’s text [68]. (For historical context of anotherfigure, Pascual Jordan, who also played a role in these developments, see, e.g., [34].)The two observables in which Born, Heisenberg, and Dirac were especially inter-ested were the position and momentum of a subatomic particle, and they employedNiels Bohr’s notion of orbits to keep track of these quantities. Mathematically, thisboils down to using matrices in order to book-keep data corresponding to the observ-ables under investigation, thus initiating matrix mechanics . The surprising outcomeof using this new matrix framework in studying subatomic particles was stated suc-cinctly as follows [33]:
The more precisely the position is determined, the less precisely the momentum is known,and vice versa.– Heisenberg’s “Uncertainty Principle”, 1927n Invitation to Noncommutative Algebra 11
More precisely, suppose that P and Q are square matrices of the same size repre-senting the observables momentum and position, respectively. The fact that P and Q do not commute typically (as one expects in classic mechanics) led to the discoveryof what Born dubbed as The Fundamental Equation of Quantum Mechanics : PQ − QP = i ¯ h ∗ I , (3)Here i is the square root of −
1, ¯ h is Planck’s constant, and i ¯ h ∗ I is the scalar mul-tiple of the identity matrix I of the same size as P and Q . For physical reasons, itwas known early in the theory of quantum mechanics that matrices P and Q thatsatisfy Equation (3) should be of infinite size, and we will recall a well-known,mathematical proof of this fact later in Proposition 1.As done in practice by many physicists and mathematicians, through rescalinglet’s consider a normalized version of The Fundamental Equation, as this still cap-tures the spirit of Heisenberg’s Uncertainty Principle: PQ − QP = I , (4)Now with today’s technology, one convenient way of studying matrix solutions P and Q to Equation (4) (or to Equation (3)) is to use the theory of representationsof (associative) algebras . To see this connection, first let’s fix a field k and for ease: Standing Hypothesis . We assume in this section that k is a field of characteristic 0.Then recall from Definition 4 (and Figure 5) that a k -algebra A is a k -vector spaceequipped with the structure of a unital ring in a compatible fashion. In this case, A = ( A , + , − , ∗ , · , , ) where ( A , + , − , ∗ , ) is the k -vector space structure where + is the abelian group operation and ∗ is scalar multiplication, and ( A , · , ) is aunital ring with · denoting its multiplication. Next, we make our vague notion ofrepresentations in Definition 2 more precise in the context of k -algebras. Definition 6.
Consider the following notions.1. For a k -vector space V , the endomorphism algebra End ( V ) on V is an k -algebraconsisting of endomorphisms of V with multiplication being composition ◦ . (If V is an n -dimensional k -vector space, then End ( V ) is isomorphic to the matrixalgebra Mat n ( k ) with matrix multiplication. Here, n could be infinite.)2. A representation of an associative k -algebra A is a k -vector space V equippedwith a k -algebra homomorphism φ : A → End ( V ) ; say φ ( a ) = : φ a ∈ End ( V ) for a ∈ A . Namely, for all a , b ∈ A , λ ∈ k , and v ∈ V , we get that φ a + b ( v ) = φ a ( v ) + φ b ( v ) , φ λ ∗ a ( v ) = λ ∗ φ a ( v ) , φ ab ( v ) = ( φ a ◦ φ b )( v ) .
3. The dimension of a representation ( V , φ ) of an associative k -algebra A is the k -vector space dimension of V , which could be infinite.Representations of associative k -algebras A go hand-in-hand with A -modules, asillustrated in Figure 12 below. k -algebras. Now for the purposes of finding matrix solutions of Equation (4), consider the k -algebra defined below. Definition 7.
The (first) Weyl algebra over a field k is the k -algebra A ( k ) generatedby noncommuting variables x and y , subject to relation yx − xy =
1. That is, A ( k ) has a k -algebra presentation A ( k ) = k (cid:104) x , y (cid:105) / ( yx − xy − ) , given as the quotient algebra of the free algebra k (cid:104) x , y (cid:105) (consisting of words invariables x and y ) by the ideal ( yx − xy − ) of k (cid:104) x , y (cid:105) . (This algebra is sometimesreferred to as the Heisenberg-Weyl algebra due to its roots in physics.)The Weyl algebra is also the first example of an algebra of differential operators –its generators x and y can be viewed as the differential operators on the polynomialalgebra k [ x ] given by multiplication by x and ddx , respectively. (Check that ddx x − x ddx is indeed the identity operator on k [ x ] .)Returning to the problem of finding n -by- n matrix solutions to Normalized Fun-damental Equation (4)– this is equivalent to the task of constructing n -dimensionalrepresentations of A ( k ) as shown in Figure 13 below. In fact, this is why A ( k ) isknown as the ring of quantum mechanics . Fig. 13: Connection between matrix solutions to N.F.E. and representations of A ( k ) .n Invitation to Noncommutative Algebra 13 Next, with the toolkit of matrices handy, we obtain a well-known result on thesize of matrix solutions to (4). We need following facts about the trace of a squarematrix X (which is the sum of the diagonal entries of X ): tr ( X ± Y ) = tr ( X ) ± tr ( Y ) and tr ( XY ) = tr ( Y X ) for any X , Y ∈ Mat n ( k ) . Proposition 1.
The Normalized Fundamental Equation (4) does not admit finite ma-trix solutions, i.e., representations of A ( k ) must be infinite-dimensional.Proof. By way of contradiction, suppose that we have matrices P , Q ∈ Mat n ( k ) with0 < n < ∞ so that PQ − QP = I . Applying trace to both sides of this equation yields:0 = tr ( PQ ) − tr ( PQ ) = tr ( PQ ) − tr ( QP ) = tr ( PQ − QP ) = tr ( I ) = n , a contradiction as desired. (cid:117)(cid:116) On the other hand, the first Weyl algebra does have an infinite-dimensional rep-resentation. Take, for instance: P = and Q =
01 01 01 0... ... . (5) ...And there are many, many more!But finding explicit matrix solutions to equations is computationally difficultin general, especially when the most important representations of an algebra areinfinite-dimensional. The power of representation theory, however, is centered onits tools to address more abstract algebraic problems that are (perhaps) related tocomputational goals. For instance, representation theory may address some of thefollowing questions for a given k -algebra A , which are all quite natural: • Do representations of
A exist ? If so, what are their dimensions ? • When are two representations considered to be the same (or isomorphic )? • Are (some of) the representations of
A parametrized by a geometric object X ?Do isomorphism classes of representations correspond bijectively to points of X ?We will explore a few of these questions and further notions in Section 5.2 to-wards a research direction in Representation Theory.The representation theory of other algebras of differential operators have alsobeen key in modeling subatomic behavior. This includes Dirac’s quantum alge-bra that addresses the question of how several position observables ( Q , . . . , Q m )and momentum observables ( P , . . . , P m ) commute, generalizing Heisenberg’s Un-certainty Principle for m = m-th Weyl algebra A m ( k ) , which has k -algebra presentation: A m ( k ) = k (cid:104) x , . . . , x m , y , . . . , y m (cid:105) ( x i x j − x j x i , y i y j − y j y i , y i x j − x j y i − δ i , j ) . (6) Here, δ i , j is the Kronecker delta, and the generators x i and y i are viewed as elementsof End ( k [ x , . . . , x m ]) given resp. by multiplication by x i and partial derivation ∂∂ x i .Want more physics?? We’re in luck– the representation theory of numerous non-commutative k -algebras play a vital role in several fields of physics. Some of thesealgebras and a physical area in which they appear are listed below. Happy exploring!
Noncommutative k -Algebras Appearance in Physics Reference (Year) W - algebras Conformal Field Theory [9] (1993)
Statistical Mechanics [64] (1982)
String Theory [6] (2000)
Yang-Mills algebras
Gauge Theory [13] (2002)
Superpotential algebras
String Theory [25] (2006)
Various Enveloping Algebrasof Lie algebras * Everywhere * Too many to list!
Let’s begin here with a question mentioned in the introduction on the ties betweensymmetries [Definition 1] and deformations [Definition 3].
Question 3.
How do we best handle (i.e., axiomatize, or “make mathematical”, theconcept of) symmetries of deformations?Several answers to this question lead us to use
Hopf algebras [Definition 11].But before we give the precise definition of this structure, we point out that Hopfalgebras became prominent in mathematics in a few waves, including: its origins inAlgebraic Topology [35], role in Combinatorics [38], and abstraction in CategoryTheory [40]. One tie to Noncommutative Algebra (in the context of Question 3) firstappeared in the 1980s in statistical mechanics, especially in the
Quantum InverseScattering Method for solving quantum integrable systems. The Hopf algebras thatarose this way were coined
Quantum Groups by Vladimir Drinfel’d [22], and havebeen a key structure in Noncommutative Algebra and physics ever since.Instead of delving further into historical details, let’s now discuss (quantum) sym-metries of (deformed) algebras through concrete examples. Fix a field k , and recallfrom Figure 5 that an associative k -algebra is a k -vector space equipped with thestructure of a (unital) ring; we consider their deformations below. Definition 8.
Fix a k -algebra A . A k -algebra A def is a deformation of A if A def and A are the same as k -vector spaces, but their respective multiplication rules are notnecessarily the same. n Invitation to Noncommutative Algebra 15 Example 1.
Our running example of a k -algebra throughout this section will be the q-polynomial algebra : k q [ x , y ] = k (cid:104) x , y (cid:105) / ( yx − qxy ) , for q ∈ k × , which is the quotient algebra of the free algebra k (cid:104) x , y (cid:105) by the ideal ( yx − qxy ) .Loosely speaking, k q [ x , y ] is a q -deformation of k [ x , y ] as the former structure ‘ap-proaches’ the latter as q →
1. More explicitly, note that k q [ x , y ] and k [ x , y ] have thesame k -vector space basis { x i y j } i , j ≥ , but their multiplication rules differ for q (cid:54) = k q [ x , y ] for q (cid:54) = k [ x , y ] .For this it is enough to consider degree-preserving symmetries , i.e., invertible trans-formations that send the generators x and y to a linear combination of themselves.Namely, let V = k x ⊕ k y be the generating space of k q [ x , y ] (or k [ x , y ] with q = GL ( V ) = GL ( k ) also inducea symmetry of k q [ x , y ] , and to do so, we need to rewrite k q [ x , y ] using the notionbelow. (From now on, we need an understanding of tensor products ⊗ and a nicediscussion of this operation can be found in [15].) Definition 9.
Given a k -vector space V , the tensor algebra T ( V ) is the k -vectorspace (cid:76) i ≥ V ⊗ i where V = k , and with multiplication given by concatenation, i.e., ( v ⊗ · · · ⊗ v m )( v m + ⊗ · · · ⊗ v m + n ) = v ⊗ · · · ⊗ v m + n . Ideals I of tensor algebras T ( V ) are defined as usual, and one can define a quo-tient k -algebra given by T ( V ) / I . Example 2.
The free algebra k (cid:104) x , y (cid:105) is identified with the tensor algebra T ( V ) on the k -vector space V = k x ⊕ k y : for the forward direction insert ⊗ between variables,and conversely suppress ⊗ between variables. The q -polynomial algebra k q [ x , y ] isthen identified as the quotient algebra of T ( k x ⊕ k y ) by the ideal ( y ⊗ x − qx ⊗ y ) .Now take g ∈ GL ( V ) for V = k x ⊕ k y . We want to extend this symmetry on V toa symmetry of k q [ x , y ] identified as T ( V ) / ( y ⊗ x − qx ⊗ y ) . Let’s assume that, as inthe case for group actions, g acts on T ( V ) diagonally: g ( v ⊗ v (cid:48) ) : = g ( v ) ⊗ g ( v (cid:48) ) , ∀ v , v (cid:48) ∈ V . (7)Now the question is: When is the ideal ( y ⊗ x − qx ⊗ y ) preserved under this action?In fact it suffices to show that g ( y ⊗ x − qx ⊗ y ) = λ ( y ⊗ x − qx ⊗ y ) , for some λ ∈ k , (8)since the g -action is degree preserving. To be concrete, say g ∈ GL ( V ) is given by g ( x ) = α x + β y and g ( y ) = γ x + δ y , for some α , β , γ , δ ∈ k . (9)Then, g ( y ⊗ x − qx ⊗ y ) = [ g ( y ) ⊗ g ( x )] − q [ g ( x ) ⊗ g ( y )] , which is equal to ( − q ) αγ ( x ⊗ x ) + ( β γ − q αδ )( x ⊗ y ) + ( αδ − q β γ )( y ⊗ x ) + ( − q ) β δ ( y ⊗ y ) . Therefore, the condition (8) is satisfied: • Always, if q = • Only when α = δ = β = γ =
0, if q = − • Only when β = γ =
0, if q (cid:54) = ± λ = αδ − β γ , the determinant of g when in matrix form.)So, when we pass from the commutative polynomial algebra k [ x , y ] to its non-commutative deformation k q [ x , y ] for q (cid:54) =
1, the amount of its degree-preservingsymmetries shrinks abruptly. This is rather unsatisfying as passing “continuously”from k [ x , y ] to k q [ x , y ] does not yield a “continuous passage” between their respec-tive degree-preserving automorphism groups.We need to think beyond group actions like those in (7). In general, we wantto construct symmetries of a k -algebra T ( V ) / I by (i) considering symmetries ofthe generating space V , (ii) extending those to symmetries of T ( V ) , and then (iii)determining which symmetries in (ii) descend to T ( V ) / I . For step (i), take V to bea representation of an algebraic object H , e.g., H could be a group or a k -algebra.(We often swap back and forth between using “representations” and “modules”.)For (ii), one needs to tackle the issue of building a direct sum and tensor product of H -representations. The former is pretty straight-forward– one can always constructthe direct sum of H -representations to get another (the first guess is most likely thecorrect one!). But if we are given two vector spaces V and V that are H -modules, Question 4.
When is V ⊗ V an H -module? If H were a group G , then one can give V ⊗ V the structure of a (left) G -modulevia (7). We can extend this linearly to get that V ⊗ V is a module over a group al-gebra on G . But if H were an arbitrary algebra, then the diagonal action on V ⊗ V does not necessarily give it the structure of an H -module (as we will see in Re-mark 1). In fact, to have an action of H on V ⊗ V we first need algebra maps ∆ : H → H ⊗ H , ∆ ( h ) (cid:55)→ ∑ h ⊗ h , and ε : H → k . Here, we use the
Sweedler notation shorthand to denote elements of ∆ ( H ) . Thesemaps should be compatible in a way that is dual to the manner that the multiplicationmap m : H ⊗ H → H and unit map η : k → H of an algebra are compatible (cf. m ( η ⊗ id H ) = id H = m ( id H ⊗ η ) ). That is, after identifying k ⊗ H = H = H ⊗ k , ( ε ⊗ id H ) ◦ ∆ = id H = ( id H ⊗ ε ) ◦ ∆ . (10) Definition 10. [63, Chapter 5] An associative k -algebra H = ( H , m , η ) is a k - bialgebra if it equipped with algebra maps ∆ ( coproduct ) and ε ( counit ), so that ( H , ∆ , ε ) is a coassociative k -coalgebra with the structures ( H , m , η ) and ( H , ∆ , ε ) being compatible. In categorical language, this is the question of whether the category of H -modules (or of repre-sentations of H ) has a monoidal structure.n Invitation to Noncommutative Algebra 17 To answer Question 4: If H is a bialgebra, the H -module structure on V ⊗ V is h ( v ⊗ v ) = : ∑ h ( v ) ⊗ h ( v ) ∀ h ∈ H and v , v ∈ V . We also get that k admits the structure of a trivial H -module via h ( k ) = ε ( h ) k . Remark 1.
We cannot always use a diagonal action– sometimes a fancier coproductis needed to address Question 4. To see this, take H to be the 2-dimensional asso-ciative k -algebra k [ h ] / ( h ) (e.g., so that we are considering linear operators that arethe zero map when composed with itself). If the coproduct of H is ∆ ( h ) = h ⊗ h ,then ε ( h ) = = ε ( h ) = ε ( h ) =
1, a contradiction. To“fix” this, check that the coproduct ∆ ( h ) = h ⊗ + ⊗ h with the counit ε ( h ) = k [ h ] / ( h ) the structure of a bialgebra over k .Moreover, one may be interested (in symmetries of) an algebra with generat-ing space V ∗ , the linear dual ; this will play a role later in Section 5.2. To getthis, we want V ∗ to have the induced structure of an H -module, and we need ananti-algebra-automorphism S : H → H of H to proceed. Definition 11. [63, Chapters 6-7] A k -bialgebra H = ( H , m , η , ∆ , ε ) is a Hopf alge-bra over k if there exists anti-automorphism S : H → H ( antipode ) so that m ◦ ( S ⊗ id H ) ◦ ∆ = η ◦ ε = m ◦ ( id H ⊗ S ) ◦ ∆ . If H is a Hopf algebra with H -module V , an action of H on V ∗ can be given by [ h ( f )]( v ) = f [ S ( h )( v )] , ∀ h ∈ H , f ∈ V ∗ , v ∈ V . Fig. 14: Symmetries deforming
Examples of Hopf algebras are group alge-bras on finite groups k G , function algebras onalgebraic groups O ( G ) , and universal envelop-ing algebras of Lie algebras U ( g ) , which areall considered “classical” in the sense that theyare commutative (as an algebra, m ◦ τ = m ) or cocommutative (as a coalgebra, τ ◦ ∆ = ∆ ), for τ ( a ⊗ b ) = b ⊗ a . Indeed, these Hopf algebrascapture the actions of a group on a k -algebra byautomorphism and actions of a Lie algebra ona k -algebra by derivation. Moreover, deforma-tions (or quantized versions) of these structures provide a setting to handle defor-mations of the aforementioned symmetries (cf. Question 3); refer to [1, 36, 51, 54]for examples of Hopf algebras arising in this fashion. We also recommend the ex-cellent text on (actions of) Hopf algebras by Susan Montgomery [57].Now we summarize a few frameworks for studying (quantum) symmetries of a k -algebra A involving a group G or a Hopf algebra H . See [57] for more details. In this case, the category of H -modules is a rigid monoidal category. For more settings of quantum symmetry, see, e.g., [63, Chapter 11] for a categorical framework.8 Chelsea Walton [G-
ACT ] Group actions on A : That is, A is a G -module with G -action map G × A → A given by ( g , a ) (cid:55)→ g ( a ) satisfying g ( ab ) = g ( a ) g ( b ) and g ( A ) = A ,for all g ∈ G and a , b ∈ A .[G- GRD ] Group gradings on A : That is, A is G -graded if A = ⊕ g ∈ G A g , for A g a k -vector space, with A g · A h ⊂ A gh . When G is finite, this is equivalent to A beingacted upon by the dual group algebra ( k G ) ∗ .[H- ACT ] Hopf algebra or bialgebra actions on A : That is, A is an H -module with H -action map H × A → A given by ( h , a ) (cid:55)→ h ( a ) with h ( ab ) = ∑ h ( a ) h ( b ) and h ( A ) = ε ( h ) A , for all h ∈ H and a , b ∈ A , with ∆ ( h ) = ∑ h ⊗ h .Finally we end with an example of a Hopf algebra action on k q [ x , y ] , illustrating ascenario where Question 3 has a possible answer. For other (more general) examplesin the literature, we refer to [41, Sections IV.7 and VII.3]. Example 3. (A simplified version of [41, Theorem VII.3.3]) For ease, we take k tobe C . Also, let q be a nonzero complex number that’s not a root of unity. We aim toproduce an action of a Hopf algebra H q over C (whose structure depends on q ) onthe q -polynomial algebra C q [ x , y ] = C (cid:104) x , y (cid:105) / ( yx − qxy ) , so that • the “limit” of H q as q → H (i.e., H is eithercommutative or cocommutative and H q is a q-deformation of H ), and • the “limit” of the H q -action on C q [ x , y ] as q → H on C [ x , y ] .We begin by defining a Hopf algebra H q with algebra presentation, H q = C (cid:104) g , g − , h (cid:105) / ( gg − − , g − g − , gh − q hg ) , along with coproduct, counit, and antipode given by ∆ ( g ) = g ⊗ g , ∆ ( g − ) = g − ⊗ g − , ∆ ( h ) = ⊗ h + h ⊗ g , ε ( g ) = , ε ( g − ) = , ε ( h ) = , S ( g ) = g − , S ( g − ) = g , S ( h ) = − hg − . Next we define a q-number [ (cid:96) ] q : = q (cid:96) − q − (cid:96) q − q − for any integer (cid:96) . Now for any element p = ∑ i , j ≥ λ i j x i y j in C q [ x , y ] , the rule below gives us an action of H q on C q [ x , y ] : g ( p ) = ∑ i , j ≥ λ i j q i − j x i y j , g − ( p ) = ∑ i , j ≥ λ i j q j − i x i y j , h ( p ) = ∑ i , j ≥ λ i j [ j ] q x i + y j − . To check this, it suffices to show that (i) the relations of H q act on C q [ x , y ] by zero,and that (ii) the relation space of C q [ x , y ] is preserved under the rule above. We’llprovide some details here and leave the rest as an exercise. We compute:For (i), ( gh − q hg )( p ) = g ( ∑ λ i j [ j ] q x i + y j − ) − q h ( ∑ λ i j q i − j x i y j )= ∑ λ i j [ j ] q q i − j + x i + y j − − q ∑ λ i j [ j ] q q i − j x i + y j − = h ( yx − qxy ) = [ ( y ) h ( x ) + h ( y ) g ( x )] − qh ( xy ) = ( x )( qx ) − q ( x ) = . n Invitation to Noncommutative Algebra 19 Now the “limit” of H q as q → H = C [ x ] ⊗ CZ , the tensor product of Hopfalgebras , for Z = (cid:104) g (cid:105) (see, e.g., [63, Exercise 2.1.19]); H is both commutative andcocommutative. Also, H q = H as C -vector spaces. Moreover, as q →
1, the genera-tors g and g − (resp., h ) of H act on C q [ x , y ] as the identity (resp., by ∂∂ y ). We highlight a couple of directions for research in Noncommutative Algebra in thissection, building on the discussions of Sections 1-4. The material below could alsoserve as a topic for an undergraduate or Master’s thesis project, or as a readingcourse topic. Finding a friendly faculty (or advanced graduate student) mentor tohelp with these pursuits is a good place to start...
Continuing the discussions of Section 2 and 4, we propose the following avenue forresearch: Study of the symmetries of (algebraic structures that generalize) Hamil-ton’s quaternions H [Problem 1]. One such generalization is given below. Definition 12. [12, Section 5.4] Fix a field k with char k (cid:54) =
2, with nonzero scalars a , b ∈ k . Then a quaternion algebra Q ( a , b ) k is a k -algebra that has an underlying4-dimensional k -vector space with basis { , i , j , k } , subject to multiplication rules i = a , j = b , i j = − ji = k . Note that k = i jk = − ab , for instance.Sometimes Q ( a , b ) k is denoted by ( a , b ) k , by ( a , b ; k ) , or even by ( a , b ) if k isunderstood. The structure above extends the construction of Hamilton’s quaternions[Definition 5], namely H = Q ( − , − ) R . Moreover, split-quaternions , Q ( − , + ) R ,also appear frequently in the literature.Fun fact: A quaternion algebra is either a 4-dimensional k -division algebra [Def-inition 4], or is isomorphic to the matrix algebra M ( k ) ! (The latter is called the split case.) Also, these cases are characterized by the norm of elements Q ( a , b ) k : N ( a + a i + a j + a k ) : = a − aa − ba + aba , for a , a , a , a ∈ k . Namely, if k has characteristic not equal to 2, then Q ( a , b ) k is a division algebraprecisely when N ( a + a i + a j + a k ) = ( a , a , a , a ) = ( , , , ) [18, Proposition 5.4.3]. For instance, H = Q ( − , − ) R is a R -division algebra since N ( a + a i + a j + a k ) = a + a + a + a for a , a , a , a ∈ R , and is 0 if and only if ( a , a , a , a ) = ( , , , ) .Quaternion algebras (in the generality of Definition 12 above) have appeared pri-marily in number theory [69] [56, Chapter 5] and in the study of quadratic forms [47,Chapter III]. They have also been used in hyperbolic geometry [52] [53, Chapter 2],and in various parts of physics and engineering; see, e.g., [5] and [61]. For moredetails about their applications and structure, see [14] and the references within.Recall from Section 4 that there are several frameworks for studying symmetriesof a k -algebra, including group actions [G- ACT ], group gradings [G-
GRD ], andHopf algebra actions [H-
ACT ]. Also, the latter symmetries are considered to be quantum symmetries if H is non(co)commutative, as discussed by Figure 14. Problem 1.
Study the (quantum) symmetries of quaternion algebras. Namely, picka setting [G-
ACT ], [G-
GRD ], [H-
ACT ], a collection of structures ( G or H ) in thisclass, and classify all such symmetries of G or H on Q ( a , b ) k .Even if this problem is not addressed in full generality, a collection of ex-amples would be quite useful for the literature. For instance, a group grading of Q ( − , − ) R = H was used in recent work of Cuadra and Etingof as a counterexam-ple to show that their main result on faithful group gradings on division algebras failswhen the ground field is not algebraically closed [18, Theorem 3.1, Example 3.4].There are also other works that partially address Problem 1, such as on groupgradings [17, 58, 59] and Hopf algebra (co)actions [21, 66]. These papers also con-tain work on (quantum) symmetries of some generalizations of quaternion algebras; Problem 1 can also be posed for these generalizations of Q ( a , b ) k as well.Moreover, a second part of Problem 1 could include the study of two algebraicstructures formed by the symmetries constructed above, namely, the subalgebra of(co)invariants , and the smash product algebra (or, skew group algebra if [G- ACT ]is used). See [57] for the definitions, examples, and a discussion of various uses ofthese algebraic structures. Overall, after one gets comfortable with the terminology,such problems are computational in nature ... and fun to do!
In this section, k is a field of characteristic zero.Towards a research direction in representation theory (continuing the discus-sion in Section 3) it is natural to think further about the representations of the firstWeyl algebra A ( k ) . Since there are no finite-dimensional representations of A ( k ) [Proposition 1], what are its infinite-dimensional representations? To get one forexample, identify A ( k ) as a ring of differential operators on k [ x ] where the gener-ators x and y act as multiplication by x and by ddx , respectively. So, by fixing a basis { , x , x , x , . . . } of k [ x ] , we get the (matrix form of) the infinite-dimensional repre-sentation in (5). Producing explicit infinite-dimensional representations of A ( k ) istough in general. But there are many works on the abstract representation theory of n Invitation to Noncommutative Algebra 21 A ( k ) and of other rings of differential operators, and we recommend the student-friendly text of S.C. Coutinho on algebraic D-modules [16] for more information.Now for a concrete research problem to pursue, we suggest working with defor-mations of Weyl algebras instead, particularly those that admit finite-dimensionalrepresentations (as this is more feasible computationally). One could: Problem 2.
Examine the (explicit) representation theory of quantum Weyl algebras ( at roots of unity ) [Definition 16].Before we discuss quantum Weyl algebras, we introduce some terminology thatwill be of use later in order to make the problem above more precise. The text [23](which, again, is student-friendly) is a nice reference for more details. Definition 13.
Take a k -algebra A with a representation φ : A → Mat n ( k ) ( ∼ = End ( V )) for V = k ⊕ n .
1. We say that φ is decomposable if we can decompose V as W ⊕ W with W , W (cid:54) = φ | W k : A → End ( W k ) are representations of A for k = , φ is indecomposable .2. The representation φ is reducible if there exists a proper subspace W of V sothat φ | W : A → End ( W ) is a representation of A ; here, φ | W is called a (proper)subrepresentation of φ . If φ does not have any proper subrepresentations, then φ is irreducible ; the corresponding A -module V is said to be simple (cf. Figure 12).3. Take another representation φ (cid:48) : A → End ( V (cid:48) ) of A . We say that φ (cid:48) is equivalent (or isomorphic ) to φ if dim V = dim V (cid:48) and there exists an invertible k -linearmap ρ : V → V (cid:48) so that ρ ( φ a ( v )) = φ (cid:48) a ( ρ ( v )) for all a ∈ A and v ∈ V .Irreducible representations are indecomposable; the converse doesn’t always hold.To understand the notions above in terms of matrix solutions of equations (cf.Figure 13), take a finitely presented k -algebra A , that is, A has finitely many non-commuting variables x i as generators, and finitely many words f j ( x ) in x i as rela-tions: A = k (cid:104) x , . . . , x t (cid:105) ( f ( x ) , . . . , f r ( x )) . Let us also fix an n -dimensional representation of A , given by φ : A → Mat n ( k ) , x i (cid:55)→ X i for i = , . . . , t . Definition 14.
Retain the notation above. Suppose that we have a matrix solution X = ( X , . . . , X t ) to the system of equations f ( x ) = · · · = f r ( x ) = X i can be written as a direct sum of matrices X i , ⊕ X i , , where • X i , k ∈ Mat n k ( k ) with k = , n and n , and • X k = ( X , k , . . . , X t , k ) is a solution to f ( x ) = · · · = f r ( x ) = k = , then the matrix solution X is decomposable . Otherwise, X is indecomposable .2. For Mat n ( k ) identified as End ( V ) with V = k ⊕ n , suppose that there exists aproper subspace W of V that is stable under the action of each X i . Then we saythat X is reducible . Otherwise, X is irreducible .3. We say that another matrix solution X (cid:48) ∈ Mat n (cid:48) ( k ) × t to the system of equations f ( x ) = · · · = f r ( x ) = equivalent (or isomorphic ) to X if n = n (cid:48) and thereexists an invertible matrix P ∈ GL n ( k ) so that P X i P − = X i for all i .So two representations of A (or, two matrix solutions of { f j ( x ) = } rj = ) areequivalent precisely when they are the same up to change of basis of V = (cid:76) ti = x i .Therefore Problem 2 can be refined as follows. Precise version of Problem 2.
Classify the explicit irreducible representations ofthe quantum Weyl algebras [Definition 16], up to equivalence.Let’s define the quantum Weyl algebras now. One way of getting these algebrasis by deforming the m -th Weyl algebras A m ( k ) from (6) via the symmetry discussedbelow. (The reader may wish to skip to Definition 16 for the outcome of this chat.) Definition 15.
Fix a k -vector space V .1. A k -linear transformation c : V ⊗ V → V ⊗ V is a braiding if it satisfies thebraid relation, ( c ⊗ id V ) ◦ ( id V ⊗ c ) ◦ ( c ⊗ id V ) = ( id V ⊗ c ) ◦ ( c ⊗ id V ) ◦ ( id V ⊗ c ) as maps V ⊗ → V ⊗ .2. A braiding H : V ⊗ V → V ⊗ V is a Hecke symmetry if it satisfies the Heckecondition, ( H − q id V ⊗ V ) ◦ ( H + q − id V ⊗ V ) = V ⊗ V → V ⊗ V ,for some nonzero q ∈ k .Given a Hecke symmetry H ∈ End ( V ⊗ V ) one can form the H -symmetricalgebra S H , q ( V ) = T ( V ) / ( Image ( H − q id V ⊗ V )) . For example, when H = flip (sending x i ⊗ x j to x j ⊗ x i ) and q = S flip , ( V ) is the symmetric algebra S ( V ) on V ; this is isomorphic to the polynomial ring k [ x , . . . , x m ] for V = (cid:76) mi = k x i .Summarizing the discussion in [28], we now build a q -version of a Weyl alge-bra using a Hecke symmetry H as follows. Consider the dual vector space V ∗ and the induced k -linear map H ∗ ∈ End ( V ∗ ⊗ V ∗ ) . Then construct the algebra A H , q ( V ⊕ V ∗ ) on V ⊕ V ∗ , which is the tensor algebra T ( V ⊕ V ∗ ) subject to therelations: Image ( H − q id V ⊗ V ) , and Image ( H ∗ − q − id V ∗ ⊗ V ∗ ) , and certain rela-tions entertwining generators from V with those from V ∗ by using H . The resultingalgebra A H , q ( V ⊕ V ∗ ) is called the quantum Weyl algebra associated to H .For simplicity, we provide the presentation of A H , q ( V ⊕ V ∗ ) for the standard1-parameter Hecke symmetry given on [37, page 442] (provided in the form of anR- matrix ). Here, V = (cid:76) mi = k x i and V ∗ = (cid:76) mi = k y i with y i : = x ∗ i (linear dual of x i ). Definition 16. [37, page 442] [28, Definition 1.4] Take m ≥
2. The is an associative k -algebra A qm ( k ) with noncommuting gen-erators x , . . . , x m , y , . . . , y m subject to relations n Invitation to Noncommutative Algebra 23 x i x j = qx j x i , y i y j = q − y j y i , ∀ i < jy i x j = qx j y i , ∀ i (cid:54) = jy i x i = + q x i y i + ( q − ) ∑ j > i x j y j , ∀ i . By convention, we define A q ( k ) to be k (cid:104) x , y (cid:105) / ( yx − qxy − ) . If q is a root of unitythen we refer to these algebras as quantum Weyl algebras at a root of unity .Notice that one gets the Weyl algebras A ( k ) [Definition 7] and A m ( k ) [Equa-tion (6)] by taking the “limit” of A q ( k ) and A qm ( k ) as q →
1, respectively.Fun fact: If q is a root of unity, say of order (cid:96) , then all irreducible representationsof a quantum Weyl algebra A H , q ( V ⊕ V ∗ ) are finite-dimensional! Moreover in thiscase, the dimension of an irreducible representation is A H , q ( V ⊕ V ∗ ) is boundedabove by some positive integer N ( (cid:96) ) depending on (cid:96) , and this bound is met mostof the time. This is part of a general phenomenon for quantum k -algebras withscalar parameters– they have infinite-dimensional irreducible representations in thegeneric case, and in the root of unity case all of their irreducible representations arefinite-dimensional. Further, in the root of unity case, most irreducible representa-tions of a quantum algebra A have dimension equal to the polynomial identity (PI)degree of A (See, for instance, the informative text of Brown-Goodearl [11]). Forexample, the PI degree of A q ( k ) is equal to (cid:96) when q is a root of unity of order (cid:96) .This leads us to discussion of a partial answer to Problem 2 . Indeed, one wasachieved for A q ( k ) , for q a root of unity of order (cid:96) , in two undergraduate researchprojects directed by E. Letzter [10] and by L. Wang [31]. The explicit irreduciblematrix solutions ( X , Y ) to the equation Y X − qXY = (cid:96) -by- (cid:96) matrices.Naturally, the next case for Problem 2 is the representation theory of quantumWeyl algebras A H , q ( V ⊕ V ∗ ) , where dim k V = q is a root of unity; this shouldbuild on the partial answer above. There are a few routes one could take, such asexamining A qm ( k ) for m ≥
2, or more generally, addressing Problem 2 for multi-parameter quantum Weyl algebras as in [28, Example 2.1] [11, Definition 1.2.6].Why care? One reason is that quantum Weyl algebras have appeared in numerousworks in mathematics and physics, including Deformation Theory [27, 28, 37, 39],Knot Theory [24], Category Theory [48], Quantum mechanics and HypergeometricFunctions [65] to name a few. Therefore, any (partial) resolution to Problem 2 wouldbe a welcomed addition to the literature. So let’s have a go at this. :)
Acknowledgements
C. Walton is partially supported by the US National Science Foundation withgrants
Photo and Figure credits
Figs. 2, 5-7, 9-10, 12-13: Author. (** = from unsplash.com)Fig. 1: Tammie Allen, @tammeallen**. Fig. 8: Wikipedia, user: JP.Fig. 3: GazzaPax (flickr.com). Fig. 11: Billy Huynh, @billy huy**.Fig. 4: Karolina Szczur, @thefoxis**. Fig 14. Dan Gold, @danielcgold**.4 Chelsea Walton
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