Download Overview Andrew Jaramillo Research Statement

Andrew Jaramillo
Research Statement
Overview
My research interests are in noncommutative algebra and quantum groups, specifically quantized
coordinate rings and quantized enveloping algebras. Though a quantum group has no widely
accepted definition, the property that many quantum groups have is they are noncommutative
deformations of classical objects with some parameter q, so that when q approaches certain special
values the classical object is recovered.
In my thesis I define two noncommutative deformations of the coordinate ring of the unipotent
radical of the standard positive (respectively negative) Borel subgroup in SLn+1 . These quantized
unipotent radicals are subalgebras of the quantized coordinate ring of the positive (respectively
negative) Borel subgroup of SLn+1 . Moreover, each of these subalgebras has the property that
when q → 1 the coordinate ring of the standard positive (respectively negative) unipotent radical
is recovered. I give a presentation for these quantized unipotent radical subalgebras and show the
following:
(i) These algebras are isomorphic (Theorem 1)
(ii) These algebras are exactly the coinvariant subalgebras of certain coactions on the quantized
coordinate rings of the standard Borel subgroups (Theorem 2)
(iii) The quantized coordinate rings of the standard Borel subgroups are smash products of the
quantized unipotent radical and the quantized coordinate ring of the standard maximal torus
(Theorem 3)
(iv) These algebras are isomorphic to the quantized universal enveloping algebra of the standard
nilpotent subalgebra of sln+1 (Theorem 4)
Moreover, these results and definitions generalize to quantizations of the unipotent radicals of the
standard parabolic subgroups of SLn+1 . Finally, I extend these results to multiparameter quantum
groups as defined in [1].
For future work, I plan on generalizing these results to “nongeneric” q, that is, to cases where q is
a root of unity. Moreover, there exist quantizations for all simple enveloping algebras of Lie algebras,
and consequently quantizations for the semisimple ones as well. Therefore, since the standard Borel
subalgebra and its nilpotent radical subalgebra are examples of solvable and nilpotent Lie algebras,
respectively, completing this project is a first step towards constructing quantizations of solvable
and nilpotent Lie algberas more generally.
Quantum Groups
Quantum groups were first discovered in the 1980’s in mathematical physics while studying the
quantum inverse scattering method. Since that time there has been much work done on quantum
groups, studying various properties of such objects. Still though, there is not yet a widely accepted
axiomatic definition for them. ([1] Problem II.10.1) Nevertheless, this has not prevented many
objects being given the title of “quantum group”. The property that many of these objects share
is that they are usually noncommutative deformations (in some sense) of classical k-algebras with
a parameter q, so that when q approaches special values, we get the classical k-algebra. Thus a
quantum group is a noncommutative object that “acts like” some classical object and from the
quantum group, the classical structure is recovered.
For instance, quantum SLn+1 (defined below) is a noncommutative deformation of O(SLn+1 ),
the ring of polynomial functions (or coordinate ring) of SLn+1 . If we take a presentation for quantum
SLn+1 then as q → 1 we get a presentation for O(SLn+1 ). Hence, at least in this example, we have
retrieved the structure of O(SLn+1 ) from the quantum group. Thus, we may think of quantum
SLn+1 as a coordinate ring of a “noncommutative space” which “limits to” O(SLn+1 ). Of course
this can be made more precise (see [3]), but the basic idea is the following: Quantum SLn+1 is the
coordinate ring of a space that has “vanished” leaving behind the “shadow” of SLn+1 .
Examples and Background
Since this idea can be rather vague, we give some more technical examples to illustrate the point
as well as introduce some background that is useful when studying quantum groups.
Example 1: Quantum Matrices
The first example is quantum 2×2 matrices denoted Oq (M22 ). This is a k-algebra with the following
presentation: the generators are X11 , X12 , X21 , X22 with relations
Xij Xil = qXil Xij for j < l
Xij Xkj = qXkj Xij for i < k
Xij Xkl = Xij Xkl for i < k and j > l
Xij Xkl − Xij Xkl = (q − q −1 )Xil Xjk for i < k and j < l
Note that as q → 1, these relations simplify to the defining relations for O(M22 ), the ring of
polynomial functions on M22 .
Using the relations for Oq (M22 ) above, we may generalize our definition and thus define quantum
n × m matrices Oq (Mnm ). Specifically, we keep the same relations with generators Xij for 1 ≤ i ≤
n and 1 ≤ j ≤ m.
Example 2: Quantum SLn+1
In Oq (M22 ) there is a “quantum determinant” denoted detq which is defined analogously to the
determinant in the classical case. Namely,
detq = X11 X22 − qX12 X21
In the classical situation notice that O(SL2 ) ∼
= O(M22 )/hdet−1i. Therefore we turn this observation
around and define quantum SL2 , denoted Oq (SL2 ), to be
Oq (SL2 ) := Oq (M22 )/hdetq − 1i
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Again, note that as q → 1 then detq becomes the ordinary determinant in O(M22 ) and so we recover
O(SL2 ).
Just as above, we may generalize this definition for arbitrary n. That is, the quantum determinant, detq , may be defined in Oq (Mn+1,n+1 ) analogously to the ordinary determinant in
O(On+1,n+1 ). Then using detq define quantum SLn+1 , denoted by Oq (SLn+1 ), in the same way as
above.
Example 3: Quantized Universal Enveloping Algebra for sln+1
Finally, we give an example that is not a deformation of a coordinate ring, rather this quantum
group is a deformation of U (sl2 ), the universal enveloping algebra (UEA) for sl2 . This k-algebra,
denoted Uq (sl2 ), has the following presentation: the generators are E, F, K ±1 with relations
KEK −1 = q 2 E
KF K −1 = q −2 F
EF − F E =
K − K −1
q − q −1
Though not quite as obvious as in the previous example, it can nevertheless be shown that Uq (sl2 )
is isomorphic to an algebra so that as q → 1 we recover the algebra U (sl2 ) [6].
Again, the above example can be generalized to a quantum UEA for sln+1 denoted by Uq (sln+1 ).
It is slightly more complicated than for quantum matrices and involves taking the Cartan relations
in U (sln+1 ) and appropriately deforming. (See [1])
Hopf Algebras and Comodules
One invaluable tool when studying quantum groups is that of a Hopf algebra. This is best introduced
by example.
Consider the algebraic group SLn+1 . Taking the group structure in SLn+1 , we may encode it
in terms of set theoretic maps. That is, multiplication is a map µ : SLn+1 × SLn+1 → SLn+1 , the
identity is an inclusion η : {1} → SLn+1 , and inversion is the map i : SLn+1 → SLn+1 defined by
i(g) = g −1 . Dualizing these maps to O(SLn+1 ) we get the corresponding maps
∆ = µ∗ : O(SLn+1 ) → O(SLn+1 × SLn+1 ) ∼
= O(SLn+1 ) ⊗ O(SLn+1 )
= η ∗ : O(SLn+1 ) → k
S = i∗ : O(SLn+1 ) → O(SLn+1 )
called comultiplication, counit, and antipode, respectively. The axioms for the group structure of
SLn+1 , in terms of the maps, are then “reversed” giving us relations among the maps ∆, , and
S. Taking the natural axioms satisfied in O(SLn+1 ) by the maps µ, i, ∆,, , and S, one obtains a
Hopf algebra. (See [6] for a full list of axioms of a Hopf algebra.) Note, Oq (SLn+1 ) and Uq (sln+1 )
are Hopf algebras with ∆, , and S appropriately defined. (See [7])
Moreover, in an analogous way to how a Hopf algebra is defined above, taking the structure maps
for a left module over a ring and reversing the structure maps, we may define a left comodule. Again,
the axioms for the left module structure give corresponding relations for the comodule structure,
only reversed. A right comodule can be defined similarly.
Current Work
For the following k is a field and q ∈ k where q is nonzero and not a root of unity.
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Standard Borel and Unipotent Radicals in SLn+1
In the classical case, when studying subgroups of an algebraic group, one often begins with the
Borel subgroups as well as the unipotent radicals of such subgroups. For instance in SLn+1 the
positive (respectively negative) standard Borel subgroup B + (respectively B − ) is the subgroup consisting of all upper (respectively lower) triangular matrices in SLn+1 . The positive (respectively
negative) unipotent radical N + (respectively N − ) of B + (respectively B − ) is the subgroup of upper
(respectively lower) triangular unipotent matrices in SLn+1 .
Since both B ± are closed subvarieties of SLn+1 , it follows then that the coordinate rings O(B ± )
have the following properties:
O(B + ) ∼
= O(SLn+1 )/hXij | i > ji
O(B − ) ∼
= O(SLn+1 )/hXij | i < ji
Similarly, since N ± are closed subvarieties, for the coordinate rings O(N ± ) we have
O(N ± ) ∼
= O(B ± )/hXii − 1 | i = 1, . . . , ni
Standard Borel and Unipotent Radicals in Quantum SLn+1
Quantum B ± denoted Oq (B ± ) are defined ([9] section 6.1) analogously to how they are for the
classical case:
Oq (B + ) := Oq (SLn+1 )/hXij | i > ji
Oq (B − ) := Oq (SLn+1 )/hXij | i < ji
However, attempting to define the quantized coordinate rings Oq (N ± ) as
Oq (N ± ) = Oq (B ± )/hXii − 1 | 1 ≤ i ≤ n + 1i
∼ k ([9] remark 6.3). Though this
would not be helpful since doing so would imply that Oq (N ± ) =
may be a nice algebra to study it is not a particularly useful analogue to the classical setting.
Therefore Oq (N ± ) should be defined in another way.
There are (surprisingly) few definitions found in the literature although some authors (e.g. [2])
have defined Oq (N ± ) to be the quantum positive (respectively negative) nilpotent subalgebra of
Uq (sln+1 ), since when q = 1 we recover O(N ± ). This definition works fine but it is not easily seen
to be a “quantum subgroup” of SLn+1 .
One difficulty for Oq (B ± ) is, in contrast to the classical case, there are no “natural” quotient
algebras of Oq (B ± ) that reduce to O(N ± ) when q = 1. ([9] remark 6.3.) Nevertheless, there exist
candidate subalgebras of Oq (B ± ) that do have this property, some of which were defined in [4].
In my work there are defined two subalgebras in Oq (B + )
−1
Oq (N + ) := k Xii−1 Xij | 1 ≤ i < j ≤ n + 1
Oq (N + )0 := k Xij Xjj
|1≤i<j ≤n+1
Both of these share the property that when q → 1 we recover O(N + ). (Similarly, there are
also defined two subalgebras Oq (N − ) and Oq (N − )0 in Oq (B − ).) Using the above I have found a
presentation for these algebras and also shown the following theorem.
Theorem 1. (Jaramillo) Oq (N ± ) and Oq (N ± )0 are all isomorphic.
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Unipotent Radicals as Coinvariants
A left (respectively right) coinvariant for a left (respectively right) comodule M over a Hopf algebra
R is any element x ∈ M so that ∆M (x) = 1 ⊗ x (respectively x ⊗ 1).
Another possible way to define Oq (N ± ) is to first note that B ± is the semidirect product
T n N ± where T is the diagonal subgroup (standard maximal torus) of SLn+1 . Thus O(N ± ) is
the algebra of coinvariants using the corresponding coaction of O(T ) on O(B ± ). Therefore another
possible definition for Oq (N ± ) is as the subalgebra of coinvariants for a suitable coaction of Oq (T )
(appropriately defined) on Oq (B ± ) using the Hopf algebra structure of Oq (SLn+1 ). Using this idea
I was able to show
Theorem 2. (Jaramillo) Oq (N ± ) and Oq (N ± )0 are the algebras of coinvariants for appropriate
Oq (T ) coactions on Oq (B ± ).
Therefore using the twisted crossed product from [8] I was able to show the following:
Theorem 3. (Jaramillo) Oq (B ± ) is isomorphic to Oq (T )#Oq (N ± ) and Oq (N ± )#Oq (T ).
Borel and Nilpotent Subgroups in Uq (sln+1 )
In sln+1 there are two important Lie subalgebras one usually studies. These are the positive (respectively negative) Borel Lie subalgebra, denoted b± , and positive (respectively negative) nilpotent
Lie subalgebra, denoted n± . Passing to the universal enveloping algebra, U (sln+1 ) there are corresponding subalgebras, U (b± ) and U (n± ), which are also called the positive/negative Borel and
nilpotent subalgebras of U (sln+1 ).
In Uq (sln+1 ) we may define two analogous subalgebras. These are called the quantum positive
(respectively negative) Borel and quantum positive (respectively negative) nilpotent subalgebras of
Uq (sln+1 ) denoted by Uq (b± ) and Uq (n± ), respectively. Using the fact that a slight extension of
Uq (b± ) is isomorphic to Oq (B ± ) established in [5] I was able to show the following:
Theorem 4. (Jaramillo) Oq (N ± ) is isomorphic to Uq (n± ).
Further Generalizations
A subgroup that contains a standard Borel subgroup, B ± , in SLn+1 is called a positive (respectively
negative) standard parabolic subgroup. The standard Levi subgroup for a standard parabolic subgroup, is the subgroup of matrices in SLn+1 with entries on the ”diagonal band” and 0 otherwise.
For example, B ± is a standard parabolic subgroup and the standard Levi subgroup is the standard
maximal torus described previously.
Quantizations of the standard parabolic and Levi subalgebras defined in [9]. In my work,
corresponding definitions for the quantizations of the unipotent radicals of standard Parabolic
groups were found and presentations shown. Again, just as for the case of the Borel, there are
several possible ways to define quantized unipotent radicals that satisfy the property that as q → 1
the standard unipotent radical of a parabolic group is recovered. However, also analogous to the
above, isomorphisms can be shown among these quantized unipotent radicals. Furthermore, using
a suitable coaction of the quantum Levi subalgebra on the parabolic subgroup, these quantum
unipotent radicals can be shown to be subalgebras of coinvariants.
Finally, the above definitions and results can be further generalized to multiparameter quantum
groups and quantized enveloping algebras. (See [1] for definitions of these objects.)
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Future Work
I plan to continue investigating the structure of quantum groups, further generalizing my results.
For instance, all of the above results were shown for q not a root of unity. My expectation is that
similar results will hold, with the appropriate modifications, for q a root of unity.
Moreover, all of the above results were for quantizations of the standard Borel and parabolic
subalgebras in Lie groups/algebras of type A. There are standard ways to define quantized coordinate rings in types B, C, D, and the exceptional types, both for q not a root of unity and q a root
of unity. [1] Moreover, there are standard positive and negative Borel subgroups and positive and
negative parabolic subgroups can also defined in these instances. Hence, a future project would
explore these cases and give definitions for positive and negative standard unipotent radicals and
show analogous theorems to those already proven. We note that the Borel subalgebras and their
nilpotent radicals are examples of solvable and nilpotent Lie algebras, respectively. Therefore after
completing this project, a first step towards constructing quantizations of solvable and nilpotent
algebras more generally.
Another future project worth pursuing involves the Dixmier map for nilpotent and solvable
Lie algebras. Let h be a nilpotent or solvable Lie algebra. Briefly, the Dixmier map gives a
correspondence from the dual , h∗ , to the primitive spectrum of U (h). This map is surjective and,
after factoring out by an appropriate group action, continuous and bijective.
The standard Borel and nilpotent radical Lie algebras of sln+1 are prototypical examples of
solvable and nilpotent Lie algebras, respectively. Thus we have the following question: Can one
find a “quantum Dixmier map” at least for quantum Borel and nilpotent subalgebras in Uq (sln+1 )?
Specifically, is there a map from some parameter space to the primitive spectrum of Uq (b± ) or
Uq (n± )? If so, is it surjective? Moreover, can we quotient out by some action to have a continuous
bijection from a parameter space to the primitive spectrum of these quantum UEAs?
Finally, many of the ring theoretic properties of quantum groups may also be a topic of interest
for undergraduates, especially those looking for research experience. Quantum groups give a large
class of examples where students will be able to investigate noncommutative algebra more generally.
Therefore students with some background in commutative ring theory will readily be able to see
parallels between commutative rings and their noncommutative quantum analogues. Hence, many
of the same questions an undergraduate studies in commutative algebra can be investigated for
quantum groups. For instance one project would be finding the primitive or prime spectrum of
quantum nilpotent radicals in SLn+1 for specific cases of n.
References
[1] Brown, K. A., and Goodearl, K. Lectures on Algebraic Quantum Groups. Advanced
Courses in Mathematics CRM Barcelona. Birkhauser, Basel, 2002.
[2] Geiss, C., Leclerc, B., and Schröer, J. Cluster structures on quantum coordinate rings.
Selecta Mathematica 19, 2 (May 2013), 337–397.
[3] Goodearl, K. Semiclassical limits of quantized coordinate rings. Advances in Ring Theory
(2010), 165–204.
[4] Goodearl, K., and Lenagan, T. H. Prime ideals invariant under winding automorphisms
in quantum matrices. International Journal of Mathematics 13, 5 (July 2002), 497–532.
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[5] Joseph, A. Quantum Groups and Their Primitive Ideals. Springer-Verlag, Berlin, 1995.
[6] Kassel, C. Quantum Groups, vol. 155 of Graduate Texts in Mathematics. Springer-Verlag,
1995.
[7] Klimyk, A., and Schmudgen, K. Quantum Groups and Their Representations. Texts and
Monographs in Physics. Springer-Verlag, Berlin, 1997.
[8] Montgomery, S. Hopf Algebras and Their Actions on Rings. No. 82 in Regional Conference
Series in Mathematics. American Mathematical Society, 1993.
[9] Parshall, B., and Wang, J.-P. Quantum Linear Groups. No. 439 in Memoirs of the
American Mathematical Society. American Mathematical Society, Providence, Rhode Island,
January 1991.
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