- Email: [email protected]
Hochschild kernel for locally bounded finite-dimensional representations of a connected Lie group
Applied Mathematics and Computation 218 (2011) 1063–1066
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Hochschild kernel for locally bounded finite-dimensional representations of a connected Lie group Alexander I. Shtern ⇑ Department of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia Institute of Systems Research (VNIISI), Russian Academy of Sciences, Moscow 117312, Russia
a r t i c l e
i n f o
Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Connected Lie group Continuous finite-dimensional representation Locally bounded finite-dimensional representation Universal representation kernel Hochschild universal kernel theorem
a b s t r a c t A direct version of Hochschild’s description of the universal representation kernel for continuous linear finite-dimensional representations of connected Lie groups is presented and used to describe the intersection of kernels of locally bounded (not necessarily continuous) linear representations of a given connected Lie group. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction For a topological group G, the intersection of all kernels of the continuous finite-dimensional representations of G [4] is now referred to as the Hochschild kernel or the universal representation kernel of G and is denoted by urk (G). Our main interest is related to a similar kernel, namely, the so-called locally bounded representation kernel of G. Definition. Let G be a topological group. The locally bounded representation kernel lbrk (G) of G is defined as the intersection of all kernels of (not necessarily continuous) locally bounded finite-dimensional representations of G. In what follows, we study connected Lie groups only. For generalities concerning topological groups and Lie groups, we refer to [3] and to [15], respectively. To clarify the matter and the setting of the problems to be considered in the text, let us cite some results of [4,6,9–13]. Evidently, lbrk (G) is a normal subgroup of G which is contained in urk (G). By Hochschild [4], urk (G) a is a closed normal subgroup of G, and, by Gotô [2], Theorem 7.1, the quotient group G/urk (G) has a faithful continuous finite-dimensional representation. Thus, urk (G) is the smallest closed normal subgroup N of G such that G/N is isomorphic to a real-analytic subgroup of some full linear group. Recall that, as was also proved by Hochschild [4], if G is a semisimple Lie group, then the group urk (G) can be described as follows. Let g be the Lie algebra of G, let C be the field of complex numbers, and let gC be the complexification of g, i.e., the semisimple Lie algebra over C obtained by forming the tensor product, over the real field R, of g with C; gC ¼ gR C. Denote by SðgÞ and SðgC Þ the simply connected Lie groups whose Lie algebras are g and gC , respectively. The injection g ! gC is the differential of a uniquely determined continuous homomorphism c of SðgÞ into SðgC Þ. The kernel D of c is a discrete central subgroup of SðgÞ. Let u be the covering epimorphism of SðgÞ onto G. ⇑ Address: Department of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia. E-mail address: [email protected]. 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.01.063
1064
A.I. Shtern / Applied Mathematics and Computation 218 (2011) 1063–1066
Theorem A [4]. If G is a semisimple Lie group, then the Hochschild (or ‘‘universal representation’’) kernel urk (G) is u(D). Thus, the universal representation kernel of the semisimple connected Lie group G is the image, under the universal covering epimorphism, of the kernel of the canonical homomorphism SðgÞ ! SðgC Þ. We also recall another result of Hochschild (preserving the notation used in [4]). Let R = radG be the radical of the connected Lie group G (the largest connected solvable normal subgroup of G). The obvious relation [G, R] # [G, G] \ R helps to understand the meaning of the following Hochschild general theorem [4]. (Here and below, the symbol [M1, M2], where M1 and M2 are subsets of a group G, always stands for the group-theoretic subgroup of G generated by the commutators 1 m1 m2 m1 1 m2 ; m1 2 M 1 ; m2 2 M 2 .) Theorem B [4]. Let G be a connected Lie group and let S be a maximal semisimple analytic subgroup of G. Let A be the closure in G of the group urk (S) (see Theorem 1). Thenurk (G) is the subgroup P of G that contains A and is such that the quotient group P=A is a (unique) maximal compact subgroup B of the closure in G=A of the radical radðG=AÞ0 of the commutator subgroup ðG=AÞ0 of G=A. It should be noted that the group urk (G) is described in this theorem in two steps rather than directly, and there was no direct description of urk (G) until now. In the present paper, we give a direct description of the kernel urk (G) and use this description to find the kernel lbrk (G) for an arbitrary connected Lie group G. Recall that the group studied by Birkhoff [1] obviously satisfies the condition urk (G) = lbrk (G) = G. However, the coincidence urk (G) = lbrk (G) can fail for the groups in question. The corresponding example was given in [12] for a reductive Lie group which is a quotient Lie group of a one-dimensional central extension of the universal covering group of SLð2; RÞ. Nevertheless, it should be noted that a connected Lie group has a faithful locally bounded finite-dimensional linear representation if and only if this group is linear [10]. This example shows that the role of discontinuous finite-dimensional representations in representing a connected Lie group can be essential and, to preserve the form of the Hochschild characterization of the kernel, we need additional assumptions. In the main theorem of [11], we have somewhat weakened the conditions of the Hochschild characterization of the group urk (G) by admitting the consideration of (not necessarily continuous) locally bounded representations, whereas in [12] we have described the kernel lbrk (G) for any connected reductive Lie group. 2. A description of the Hochschild universal kernel for connected Lie groups As promised above, we begin with a direct description of the Hochschild universal kernel for connected Lie groups. It is clear that every continuous finite-dimensional linear representation is trivial on every compact subgroup of the closure ½G; R of the commutator subgroup [G, R] of G and R (and ½G; R is a Lie group because it is a closed subgroup of the Lie group G). Since G and R are connected, it follows that [G, R] and ½G; R are connected as well, and therefore every maximal compact subgroup B of the connected Lie group ½G; R is connected (Theorem 15 in [5]). Since this group belongs to urk (G), it is central in G, and therefore this group is unique, and we face the largest maximal compact subgroup B of ½G; R, where B is automatically Abelian and connected. Nevertheless, when joining this group to A, one cannot consider the closed normal subgroup generated by these two groups as a candidate for the urk (G). Let us recall the corresponding example (see [9]). Example. Consider the well-known example of a connected Lie group with a reductive Lie algebra for which the commutator subgroup is dense in the group (see, e.g., [11], Chapter 3, Exercise 47). Let G0 be the universal covering group of SLð2; RÞ and let Z be the center of G0, which is isomorphic to the group of integers, Z. Let a 2 R n Q, where Q stands for the subfield of rational numbers in R. Let T ¼ R=Z be the circle group and let D be the kernel of the continuous homomorphism
ðt; nÞ # expð2piðt þ azðnÞÞ;
t 2 R;
n2Z
of the direct product R Z onto T, where z(n) stands for the integer corresponding to n under an isomorphism between Z and Z. Then, as can readily be seen, D is a discrete subgroup of R Z such that ðR ZÞ=D is isomorphic to T, and thus D is a closed central normal subgroup of R G0 . Let G ¼ ðR G0 Þ=D, let C be the center of G, and let p be the canonical mapping taking R G0 onto the quotient group G. The immediate verification shows that
pðf0g G0 Þ ¼ ½G; G; pðR ZÞ ¼ C: Thus, the closure of [G, G] is the entire group G. However, as is well known, a connected Lie group having a faithful continuous finite-dimensional representation has a closed Levi subgroup and the closed commutator subgroup (see, e.g., [11], Chapter 3, Exercise 41). A quotient group of G can satisfy these conditions only if the corresponding normal subgroup contains the center C of G; C ¼ pðR ZÞ, and thus urk (G) = C. However, it can readily be seen that lbrk (G) coincides with the image p({0} Z) of the center of the second factor in G because the group [G, G] = p({0} G0) is obviously isomorphic to the group G0, whereas the natural complement pðR f0gÞ is commutative and automatically has a sufficient family of (possibly discontinuous) unitary characters. Thus,
urk ðG0 Þ ¼ lbrk ðGÞ ( urk ðG0 Þ ¼ C;
A.I. Shtern / Applied Mathematics and Computation 218 (2011) 1063–1066
1065
as was to be proved. (Recall that the center of the semisimple part [G, G] of G belongs not only to [G, G] but also to the radical C ¼ pðR f0gÞ of G.) This example shows that the correct form of a direct description of the kernel in question is more subtle. Theorem 1. Let G be a connected Lie group and let S be a maximal semisimple analytic subgroup of G. Let A be the closure in G of the group urk (S) (see Theorem 1 ). Then urk (G) is the central subgroup P of G that contains A and satisfies the following condition: if the group A (which is a compactly generated Lie group) is represented (as in Section 29 of [16] , in combination with the classical theorem on the structure of finitely generated Abelian groups) as the direct product
A¼FLQ V
ð1Þ
of a finite Abelian group F, a finite-dimensional lattice L, a compact connected Abelian group Q, and the additive group V of a finitedimensional vector space (which we still denote by V), then the group P is obtained from A by replacing the group Q by the group B (introduced above as the largest maximal compact subgroup of ½G; R, which is automatically Abelian and connected) and by replacing every subgroup ‘of the lattice L entirely contained in some central vector subgroup of ½G; R by the vector subgroup L generated by this lattice. Proof of Theorem 1. Denote the group defined according to the construction in the theorem by P. We are to prove that P ¼ P. Due to a lack of the room, we only sketch the proof. Note that the group A is a subgroup of a semisimple Lie group, and hence A [G, G]. Therefore, A ½G; G. Since the connected component of the Abelian group A belongs to R, it follows that this component belongs to the intersection ½G; G \ R. However, it follows from a remark in [13] that, within compact sets of G, every element of the intersection ½G; G \ R is a product of an element of S \ R and an element of ½G; R. This yields that the quotient group P=A entirely belongs to the closed connected subgroup ½G=A; RA=A. Note that the group P has a compact connected image in the quotient group G=A (because the quotient group L=‘ is compact and connected and the image of the compact connected group B is compact and connected as well). Since P is obtained from A by using additional generators forming connected subgroups of ½G; R, it follows that P P. Let us represent the Abelian Lie group P in the form of the product
P ¼ F P LP Q P V P ;
ð2Þ
where FP is finite, LP is a finite-dimensional lattice, QP is a compact connected Abelian group, and VP is the additive group of a vector space. This group contains A (of the form F L Q V) in such a way that P=A is a compact connected group. This means that V VP, and there is a lattice L \ V spanning a vector subspace of VP complementing V. Since the connected component of the quotient group P=A is the entire quotient group, it follows that the part of the lattice LP which is outside L \ V and the finite group FP, i.e., all elements well defining different connected components in P, have representatives in L and F modulo the component of A. This means that the only modifications of the group A occurring during the transformation to P are the passages to R from T and the occurring of new tori T as factors of the component. (Certainly, the compact part of the enlarged connected component of P can contain a subgroup of F; this modifies F but does not influence the structure of P.) By the very Hochschild theorem, the reconstruction of the connected component of A into the connected component of P happens within the center of G, which completes the Proof of Theorem 1. h The main point of importance for us is that no new free Abelian groups occur when passing from (1) to (2), namely, L has representatives for all elements of LP. Corollary 1. In particular, the relation urk (G) = lbrk (G) certainly holds for a connected Lie group G if the group G is perfect (i.e., if G = G0 ).
3. A description of the locally bounded universal kernel for connected Lie groups The author of the present paper has recently obtained a theorem on the automatic continuity of finite-dimensional locally bounded representations of Lie groups generalizing the famous van der Waerden theorem [14] and claiming that every finite-dimensional locally bounded representation of a connected Lie group is continuous on the commutator subgroup [G, G] of G ([7–9]). This theorem shows that, within the commutator subgroup of G, the difference between urk (G) and lbrk (G) is mainly related to extension problems for finite-dimensional representations (from the commutator subgroup of to the entire group). This theorem is needed in the following description of the group lbrk (G). Theorem 2. Let G be a connected Lie group and let P be the universal representation kernel urk (G) of G. Let [G, G] be the commutator subgroup of G. The locally bounded representation kernel lbrk (G) of G is the intersection of P = urk (G) with the commutator subgroup [G, G].
1066
A.I. Shtern / Applied Mathematics and Computation 218 (2011) 1063–1066
Proof of Theorem 2. Let G be a connected Lie group. Let G = SR be a Levi decomposition of G, where S stands for a Levi subgroup and R for the radical of G. Let S[G, R] be the group-theoretic subgroup of G generated by the elements of S and the commutators of the form grg1r1, g 2 G, r 2 R. Since R is a normal subgroup of G, it follows that every element g 2 G is a product of the form g = sr, s 2 S, r 2 R. This readily implies that the (group-theoretic) subgroup S[G, R] is a (group-theoretic) normal subgroup of G. Since every commutator in G is also a product of an element of S, which is a commutator in turn, and an element of R, which is clearly a product of an element of S \ R and an element of [G, R], it follows that the commutator subgroup [G, G] coincides with S[G, R]. Since the quotient group G/[G, G] is Abelian, it follows that this group has a separating family of unitary characters, which are automatically locally bounded when regarded as (one-dimensional) representations of G. Thus, if two elements of G have different images modulo [G, G], then they are distinguished by locally bounded one-dimensional representations of G. Moreover, by the very definition of urk (G), any element of [G, G] not belonging to P admits a continuous finite-dimensional representation of G taking this element of [G, G] to a nonidentity matrix; this means that the kernel lbrk (G) is contained in P \ [G, G]. Conversely, every locally bounded finite-dimensional representation p of G is continuous on [G, G] by the theorem cited before the statement of Theorem 2. This shows that the restriction of p to the subgroup A is trivial, whereas the closure A \ ½G; G of A in [G, G] is contained in lbrk (G). It can be proved (using properties of the Weyl group of S) that, if g = sr 2 lbrk (G), then s 2 A, which enables us to reduce the problem concerning lbrk (G) to the intersection P \ [G, S]. Using compactness arguments and the fact that the closure A of A has bounded image in every locally bounded (not necessarily continuous) finite-dimensional representation of G, one can see that the entire complete preimage P of the maximal compact subgroup B of the closure in G=A of the radical rad ðG=AÞ0 of the commutator subgroup ðG=AÞ0 of G=A (see Theorem B above) has bounded image in every locally bounded finite-dimensional representation of G as well. On the other hand, repeating (with minor modifications) the proof of the analog of the Lie theorem for non necessarily continuous representations of solvable Lie groups (see [7–9]), one can show that the image of [G, R] in any locally bounded finite-dimensional representation belongs to the family of unipotent matrices. This shows that the intersection P \ [G, R] belongs to lbrk (G), which completes the proof of the theorem. h 4. Congratulations I am very glad to send my warmest congratulations and best wishes to Professor Hari Srivastava on the occasion of the beautiful date of life. Acknowledgements The research was partially supported by the Russian Foundation for Basic Research under Grant No. 08-01-00034 and by the Program of Supporting Leading Scientific Schools under Grant No. NSh-1562.2008.1. References [1] G. Birkhoff, Lie groups simply isomorphic with no linear group, Bull. Amer. Math. Soc. 42 (1936) 883–888. [2] M. Gotô, Faithful representations of lie groups. I, II, Math. Japonicae 1 (1948) 107–119; M. Gotô, Nagoya Math. J. 1 (1950) 91–107. [3] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. I, Springer-Verlag, Berlin, New York, 1979. [4] G.P. Hochschild, The universal representation kernel of a lie group, Proc. Amer. Math. Soc. 11 (1960) 625–629. [5] K. Iwasawa, On some types of topological groups, Ann. Math. 50 (1949) 507–558. [6] M. Moskowitz, Faithful representations and a local property of lie groups, Math. Z. 143 (1975) 193–198. [7] A.I. Shtern, Van der Waerden’s continuity theorem for the commutator subgroups of connected lie groups and Mishchenko’s conjecture, Adv. Stud. Contemp. Math. (Kyungshang) 13 (2) (2006) 143–158. [8] A.I. Shtern, Finite-dimensional quasirepresentations of connected lie groups and Mishchenko’s conjecture, Fundam. Prikl. Mat. 13 (7) (2007) 85–225; A.I. Shtern, J. Math. Sci. (N.Y.) (English translation) 159 (5) (2009) 653–751. [9] A.I. Shtern, A version of the van der Waerden theorem and the proof of the Mishchenko conjecture for homomorphisms of locally compact groups, Izv. Ross. Akad. Nauk Ser. Mat. 72 (1) (2008) 183–224; A.I. Shtern, Izv. Math. (English translation) 72 (1) (2008) 169–205. [10] A.I. Shtern, Connected lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations, Russ. J. Math. Phys. 16 (4) (2009) 566–567. [11] A.I. Shtern, Applications of automatic continuity results to analogs of the Freudenthal–Weil and Hochschild theorems, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2) (2010) 203–212. [12] A.I. Shtern, Hochschild kernel for locally bounded finite-dimensional representations of a connected reductive lie group, Proc. Jangjeon Math. Soc. 13 (2) (2010) 127–132. [13] A.I. Shtern, Connected locally compact groups having sufficiently many finite-dimensional linear representations, Adv. Stud. Contemp. Math. (Kyungshang) 21 (1) (2011) 95–102. [14] B.L. van der Waerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Z. 36 (1933) 780–786. [15] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1974. [16] A. Weil, L’intégration dans les groupes topologiques et ses applications, Act. Sc. Ind., second ed., vol. 1145, Hermann, Paris, 1953.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Hochschild kernel for locally bounded finite-dimensional representations of a connected Lie group Alexander I. Shtern ⇑ Department of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia Institute of Systems Research (VNIISI), Russian Academy of Sciences, Moscow 117312, Russia
a r t i c l e
i n f o
Dedicated to Professor H. M. Srivastava on the Occasion of his Seventieth Birth Anniversary Keywords: Connected Lie group Continuous finite-dimensional representation Locally bounded finite-dimensional representation Universal representation kernel Hochschild universal kernel theorem
a b s t r a c t A direct version of Hochschild’s description of the universal representation kernel for continuous linear finite-dimensional representations of connected Lie groups is presented and used to describe the intersection of kernels of locally bounded (not necessarily continuous) linear representations of a given connected Lie group. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction For a topological group G, the intersection of all kernels of the continuous finite-dimensional representations of G [4] is now referred to as the Hochschild kernel or the universal representation kernel of G and is denoted by urk (G). Our main interest is related to a similar kernel, namely, the so-called locally bounded representation kernel of G. Definition. Let G be a topological group. The locally bounded representation kernel lbrk (G) of G is defined as the intersection of all kernels of (not necessarily continuous) locally bounded finite-dimensional representations of G. In what follows, we study connected Lie groups only. For generalities concerning topological groups and Lie groups, we refer to [3] and to [15], respectively. To clarify the matter and the setting of the problems to be considered in the text, let us cite some results of [4,6,9–13]. Evidently, lbrk (G) is a normal subgroup of G which is contained in urk (G). By Hochschild [4], urk (G) a is a closed normal subgroup of G, and, by Gotô [2], Theorem 7.1, the quotient group G/urk (G) has a faithful continuous finite-dimensional representation. Thus, urk (G) is the smallest closed normal subgroup N of G such that G/N is isomorphic to a real-analytic subgroup of some full linear group. Recall that, as was also proved by Hochschild [4], if G is a semisimple Lie group, then the group urk (G) can be described as follows. Let g be the Lie algebra of G, let C be the field of complex numbers, and let gC be the complexification of g, i.e., the semisimple Lie algebra over C obtained by forming the tensor product, over the real field R, of g with C; gC ¼ gR C. Denote by SðgÞ and SðgC Þ the simply connected Lie groups whose Lie algebras are g and gC , respectively. The injection g ! gC is the differential of a uniquely determined continuous homomorphism c of SðgÞ into SðgC Þ. The kernel D of c is a discrete central subgroup of SðgÞ. Let u be the covering epimorphism of SðgÞ onto G. ⇑ Address: Department of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia. E-mail address: [email protected]. 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.01.063
1064
A.I. Shtern / Applied Mathematics and Computation 218 (2011) 1063–1066
Theorem A [4]. If G is a semisimple Lie group, then the Hochschild (or ‘‘universal representation’’) kernel urk (G) is u(D). Thus, the universal representation kernel of the semisimple connected Lie group G is the image, under the universal covering epimorphism, of the kernel of the canonical homomorphism SðgÞ ! SðgC Þ. We also recall another result of Hochschild (preserving the notation used in [4]). Let R = radG be the radical of the connected Lie group G (the largest connected solvable normal subgroup of G). The obvious relation [G, R] # [G, G] \ R helps to understand the meaning of the following Hochschild general theorem [4]. (Here and below, the symbol [M1, M2], where M1 and M2 are subsets of a group G, always stands for the group-theoretic subgroup of G generated by the commutators 1 m1 m2 m1 1 m2 ; m1 2 M 1 ; m2 2 M 2 .) Theorem B [4]. Let G be a connected Lie group and let S be a maximal semisimple analytic subgroup of G. Let A be the closure in G of the group urk (S) (see Theorem 1). Thenurk (G) is the subgroup P of G that contains A and is such that the quotient group P=A is a (unique) maximal compact subgroup B of the closure in G=A of the radical radðG=AÞ0 of the commutator subgroup ðG=AÞ0 of G=A. It should be noted that the group urk (G) is described in this theorem in two steps rather than directly, and there was no direct description of urk (G) until now. In the present paper, we give a direct description of the kernel urk (G) and use this description to find the kernel lbrk (G) for an arbitrary connected Lie group G. Recall that the group studied by Birkhoff [1] obviously satisfies the condition urk (G) = lbrk (G) = G. However, the coincidence urk (G) = lbrk (G) can fail for the groups in question. The corresponding example was given in [12] for a reductive Lie group which is a quotient Lie group of a one-dimensional central extension of the universal covering group of SLð2; RÞ. Nevertheless, it should be noted that a connected Lie group has a faithful locally bounded finite-dimensional linear representation if and only if this group is linear [10]. This example shows that the role of discontinuous finite-dimensional representations in representing a connected Lie group can be essential and, to preserve the form of the Hochschild characterization of the kernel, we need additional assumptions. In the main theorem of [11], we have somewhat weakened the conditions of the Hochschild characterization of the group urk (G) by admitting the consideration of (not necessarily continuous) locally bounded representations, whereas in [12] we have described the kernel lbrk (G) for any connected reductive Lie group. 2. A description of the Hochschild universal kernel for connected Lie groups As promised above, we begin with a direct description of the Hochschild universal kernel for connected Lie groups. It is clear that every continuous finite-dimensional linear representation is trivial on every compact subgroup of the closure ½G; R of the commutator subgroup [G, R] of G and R (and ½G; R is a Lie group because it is a closed subgroup of the Lie group G). Since G and R are connected, it follows that [G, R] and ½G; R are connected as well, and therefore every maximal compact subgroup B of the connected Lie group ½G; R is connected (Theorem 15 in [5]). Since this group belongs to urk (G), it is central in G, and therefore this group is unique, and we face the largest maximal compact subgroup B of ½G; R, where B is automatically Abelian and connected. Nevertheless, when joining this group to A, one cannot consider the closed normal subgroup generated by these two groups as a candidate for the urk (G). Let us recall the corresponding example (see [9]). Example. Consider the well-known example of a connected Lie group with a reductive Lie algebra for which the commutator subgroup is dense in the group (see, e.g., [11], Chapter 3, Exercise 47). Let G0 be the universal covering group of SLð2; RÞ and let Z be the center of G0, which is isomorphic to the group of integers, Z. Let a 2 R n Q, where Q stands for the subfield of rational numbers in R. Let T ¼ R=Z be the circle group and let D be the kernel of the continuous homomorphism
ðt; nÞ # expð2piðt þ azðnÞÞ;
t 2 R;
n2Z
of the direct product R Z onto T, where z(n) stands for the integer corresponding to n under an isomorphism between Z and Z. Then, as can readily be seen, D is a discrete subgroup of R Z such that ðR ZÞ=D is isomorphic to T, and thus D is a closed central normal subgroup of R G0 . Let G ¼ ðR G0 Þ=D, let C be the center of G, and let p be the canonical mapping taking R G0 onto the quotient group G. The immediate verification shows that
pðf0g G0 Þ ¼ ½G; G; pðR ZÞ ¼ C: Thus, the closure of [G, G] is the entire group G. However, as is well known, a connected Lie group having a faithful continuous finite-dimensional representation has a closed Levi subgroup and the closed commutator subgroup (see, e.g., [11], Chapter 3, Exercise 41). A quotient group of G can satisfy these conditions only if the corresponding normal subgroup contains the center C of G; C ¼ pðR ZÞ, and thus urk (G) = C. However, it can readily be seen that lbrk (G) coincides with the image p({0} Z) of the center of the second factor in G because the group [G, G] = p({0} G0) is obviously isomorphic to the group G0, whereas the natural complement pðR f0gÞ is commutative and automatically has a sufficient family of (possibly discontinuous) unitary characters. Thus,
urk ðG0 Þ ¼ lbrk ðGÞ ( urk ðG0 Þ ¼ C;
A.I. Shtern / Applied Mathematics and Computation 218 (2011) 1063–1066
1065
as was to be proved. (Recall that the center of the semisimple part [G, G] of G belongs not only to [G, G] but also to the radical C ¼ pðR f0gÞ of G.) This example shows that the correct form of a direct description of the kernel in question is more subtle. Theorem 1. Let G be a connected Lie group and let S be a maximal semisimple analytic subgroup of G. Let A be the closure in G of the group urk (S) (see Theorem 1 ). Then urk (G) is the central subgroup P of G that contains A and satisfies the following condition: if the group A (which is a compactly generated Lie group) is represented (as in Section 29 of [16] , in combination with the classical theorem on the structure of finitely generated Abelian groups) as the direct product
A¼FLQ V
ð1Þ
of a finite Abelian group F, a finite-dimensional lattice L, a compact connected Abelian group Q, and the additive group V of a finitedimensional vector space (which we still denote by V), then the group P is obtained from A by replacing the group Q by the group B (introduced above as the largest maximal compact subgroup of ½G; R, which is automatically Abelian and connected) and by replacing every subgroup ‘of the lattice L entirely contained in some central vector subgroup of ½G; R by the vector subgroup L generated by this lattice. Proof of Theorem 1. Denote the group defined according to the construction in the theorem by P. We are to prove that P ¼ P. Due to a lack of the room, we only sketch the proof. Note that the group A is a subgroup of a semisimple Lie group, and hence A [G, G]. Therefore, A ½G; G. Since the connected component of the Abelian group A belongs to R, it follows that this component belongs to the intersection ½G; G \ R. However, it follows from a remark in [13] that, within compact sets of G, every element of the intersection ½G; G \ R is a product of an element of S \ R and an element of ½G; R. This yields that the quotient group P=A entirely belongs to the closed connected subgroup ½G=A; RA=A. Note that the group P has a compact connected image in the quotient group G=A (because the quotient group L=‘ is compact and connected and the image of the compact connected group B is compact and connected as well). Since P is obtained from A by using additional generators forming connected subgroups of ½G; R, it follows that P P. Let us represent the Abelian Lie group P in the form of the product
P ¼ F P LP Q P V P ;
ð2Þ
where FP is finite, LP is a finite-dimensional lattice, QP is a compact connected Abelian group, and VP is the additive group of a vector space. This group contains A (of the form F L Q V) in such a way that P=A is a compact connected group. This means that V VP, and there is a lattice L \ V spanning a vector subspace of VP complementing V. Since the connected component of the quotient group P=A is the entire quotient group, it follows that the part of the lattice LP which is outside L \ V and the finite group FP, i.e., all elements well defining different connected components in P, have representatives in L and F modulo the component of A. This means that the only modifications of the group A occurring during the transformation to P are the passages to R from T and the occurring of new tori T as factors of the component. (Certainly, the compact part of the enlarged connected component of P can contain a subgroup of F; this modifies F but does not influence the structure of P.) By the very Hochschild theorem, the reconstruction of the connected component of A into the connected component of P happens within the center of G, which completes the Proof of Theorem 1. h The main point of importance for us is that no new free Abelian groups occur when passing from (1) to (2), namely, L has representatives for all elements of LP. Corollary 1. In particular, the relation urk (G) = lbrk (G) certainly holds for a connected Lie group G if the group G is perfect (i.e., if G = G0 ).
3. A description of the locally bounded universal kernel for connected Lie groups The author of the present paper has recently obtained a theorem on the automatic continuity of finite-dimensional locally bounded representations of Lie groups generalizing the famous van der Waerden theorem [14] and claiming that every finite-dimensional locally bounded representation of a connected Lie group is continuous on the commutator subgroup [G, G] of G ([7–9]). This theorem shows that, within the commutator subgroup of G, the difference between urk (G) and lbrk (G) is mainly related to extension problems for finite-dimensional representations (from the commutator subgroup of to the entire group). This theorem is needed in the following description of the group lbrk (G). Theorem 2. Let G be a connected Lie group and let P be the universal representation kernel urk (G) of G. Let [G, G] be the commutator subgroup of G. The locally bounded representation kernel lbrk (G) of G is the intersection of P = urk (G) with the commutator subgroup [G, G].
1066
A.I. Shtern / Applied Mathematics and Computation 218 (2011) 1063–1066
Proof of Theorem 2. Let G be a connected Lie group. Let G = SR be a Levi decomposition of G, where S stands for a Levi subgroup and R for the radical of G. Let S[G, R] be the group-theoretic subgroup of G generated by the elements of S and the commutators of the form grg1r1, g 2 G, r 2 R. Since R is a normal subgroup of G, it follows that every element g 2 G is a product of the form g = sr, s 2 S, r 2 R. This readily implies that the (group-theoretic) subgroup S[G, R] is a (group-theoretic) normal subgroup of G. Since every commutator in G is also a product of an element of S, which is a commutator in turn, and an element of R, which is clearly a product of an element of S \ R and an element of [G, R], it follows that the commutator subgroup [G, G] coincides with S[G, R]. Since the quotient group G/[G, G] is Abelian, it follows that this group has a separating family of unitary characters, which are automatically locally bounded when regarded as (one-dimensional) representations of G. Thus, if two elements of G have different images modulo [G, G], then they are distinguished by locally bounded one-dimensional representations of G. Moreover, by the very definition of urk (G), any element of [G, G] not belonging to P admits a continuous finite-dimensional representation of G taking this element of [G, G] to a nonidentity matrix; this means that the kernel lbrk (G) is contained in P \ [G, G]. Conversely, every locally bounded finite-dimensional representation p of G is continuous on [G, G] by the theorem cited before the statement of Theorem 2. This shows that the restriction of p to the subgroup A is trivial, whereas the closure A \ ½G; G of A in [G, G] is contained in lbrk (G). It can be proved (using properties of the Weyl group of S) that, if g = sr 2 lbrk (G), then s 2 A, which enables us to reduce the problem concerning lbrk (G) to the intersection P \ [G, S]. Using compactness arguments and the fact that the closure A of A has bounded image in every locally bounded (not necessarily continuous) finite-dimensional representation of G, one can see that the entire complete preimage P of the maximal compact subgroup B of the closure in G=A of the radical rad ðG=AÞ0 of the commutator subgroup ðG=AÞ0 of G=A (see Theorem B above) has bounded image in every locally bounded finite-dimensional representation of G as well. On the other hand, repeating (with minor modifications) the proof of the analog of the Lie theorem for non necessarily continuous representations of solvable Lie groups (see [7–9]), one can show that the image of [G, R] in any locally bounded finite-dimensional representation belongs to the family of unipotent matrices. This shows that the intersection P \ [G, R] belongs to lbrk (G), which completes the proof of the theorem. h 4. Congratulations I am very glad to send my warmest congratulations and best wishes to Professor Hari Srivastava on the occasion of the beautiful date of life. Acknowledgements The research was partially supported by the Russian Foundation for Basic Research under Grant No. 08-01-00034 and by the Program of Supporting Leading Scientific Schools under Grant No. NSh-1562.2008.1. References [1] G. Birkhoff, Lie groups simply isomorphic with no linear group, Bull. Amer. Math. Soc. 42 (1936) 883–888. [2] M. Gotô, Faithful representations of lie groups. I, II, Math. Japonicae 1 (1948) 107–119; M. Gotô, Nagoya Math. J. 1 (1950) 91–107. [3] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. I, Springer-Verlag, Berlin, New York, 1979. [4] G.P. Hochschild, The universal representation kernel of a lie group, Proc. Amer. Math. Soc. 11 (1960) 625–629. [5] K. Iwasawa, On some types of topological groups, Ann. Math. 50 (1949) 507–558. [6] M. Moskowitz, Faithful representations and a local property of lie groups, Math. Z. 143 (1975) 193–198. [7] A.I. Shtern, Van der Waerden’s continuity theorem for the commutator subgroups of connected lie groups and Mishchenko’s conjecture, Adv. Stud. Contemp. Math. (Kyungshang) 13 (2) (2006) 143–158. [8] A.I. Shtern, Finite-dimensional quasirepresentations of connected lie groups and Mishchenko’s conjecture, Fundam. Prikl. Mat. 13 (7) (2007) 85–225; A.I. Shtern, J. Math. Sci. (N.Y.) (English translation) 159 (5) (2009) 653–751. [9] A.I. Shtern, A version of the van der Waerden theorem and the proof of the Mishchenko conjecture for homomorphisms of locally compact groups, Izv. Ross. Akad. Nauk Ser. Mat. 72 (1) (2008) 183–224; A.I. Shtern, Izv. Math. (English translation) 72 (1) (2008) 169–205. [10] A.I. Shtern, Connected lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations, Russ. J. Math. Phys. 16 (4) (2009) 566–567. [11] A.I. Shtern, Applications of automatic continuity results to analogs of the Freudenthal–Weil and Hochschild theorems, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2) (2010) 203–212. [12] A.I. Shtern, Hochschild kernel for locally bounded finite-dimensional representations of a connected reductive lie group, Proc. Jangjeon Math. Soc. 13 (2) (2010) 127–132. [13] A.I. Shtern, Connected locally compact groups having sufficiently many finite-dimensional linear representations, Adv. Stud. Contemp. Math. (Kyungshang) 21 (1) (2011) 95–102. [14] B.L. van der Waerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Z. 36 (1933) 780–786. [15] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1974. [16] A. Weil, L’intégration dans les groupes topologiques et ses applications, Act. Sc. Ind., second ed., vol. 1145, Hermann, Paris, 1953.