Seminar Sophus Lie 3 (1993) 221–227 Version of July 15, 1995

On Algebraic and Semialgebraic Groups and Semigroups

Helmut Boseck

0 . This is a “semi”lecture^{1} whose background is as follows: The Lie-theory of subsemigroups of a Liegroup, as it is considered in the compact presentation by K. H. Hofmann in volume 1, no. 1 of this seminar [4], largely reduces to a Lie-theory of subgroups in the algebraic case. Specifically, if the group G in question and the subsemigroup S are both affine algebraic varieties the S is in fact a subgroup. This fact is well known in the theory of algebraic semigroups and a consequence of a theorem on subsemigroups of algebraic Lie groups published by C. Chevalley in the second volume of his book on Lie groups in 1951 [2]. We shall rewrite the theorem using Hopf algebras and consider several consequences. The second part of the lecture deals with the main consequence. It answers the question which class of subsemigroups of an algebraic group one should consider in order to develop a “satisfactory” Lie-theory of semigroups in algebraic Lie groups by methods close to algebraic geometry. In the case of a real base field we propose the consideration of semialgebraic semigroups and illustrate this proposal by some basic facts and interesting examples as well.

1. The CHEVALLEY theorem and its consequences

Theorem 1.1. Let 𝕂 denote an infinite base field, and P be the Hopf algebra of polynomial functions on GL(n; 𝕂). Assume that S ⊆ GL(n; 𝕂) is a subsemigroup, and 𝔞 = ann_{P} S its annihilator ideal in the algebra P. Then we have the following conclusions

i) 𝔞 is a coideal in P, too, i.e., Δ𝔞 ⊆ 𝔞 ⊗ P + P ⊗ 𝔞;

ii) G(𝔞) = {g ∈ G(P) : (id_{P} ⊗g)Δ𝔞 ⊆ 𝔞} is a group;

iii) G = ann_{GL(n;𝕂)}𝔞 is an affine algebraic group canonically isomorphic to G(𝔞), and 𝔞 = ann_{P}G.

By Δ we denote the coproduct of the Hopf algebra P, while G(P) denotes the structure group of P. In other words, G(P) is the set of homomorphisms of the commutative algebra P onto 𝕂 with convolution as multiplication. The canonical isomorphism between G and G(𝔞) is the restriction of the canonical isomorphism which maps every g ∈ GL(n; 𝕂) onto the homomorphism P → 𝕂 defined by evaluation at g. A sketchy proof of ii) and iii) runs as follows: The mapping g ↦ r(g) with r(g) = (id_{P} ⊗g)Δ ∈ Aut P is just the so-called right regular representation of G(P) on the linear space P. Now ii) follows from Lemma 1 of Chevalley stating: if 𝔞 is a linear subspace of P and r(g)𝔞 ⊆ 𝔞 for every g ∈ G(P) ,then r(g)𝔞 = 𝔞. If G′(𝔞) = ann_{G(P)} 𝔞, then evidently we have G′(𝔞) ⊆ G(𝔞), for r(g)𝔞 = (id_{P} ⊗g)Δ𝔞 ⊆ 𝔞g(P) = 𝔞. Assume g_{0} ∈ G′(𝔞). Then once more by Lemma 1 we get r(g_{0})𝔞 = 𝔞, and for every p ∈ 𝔞 there exists a q ∈ 𝔞 such that p = r(g_{0})q. Let ε ∈ G(P) denote the identity, i.e., ε is the counit of P, then it holds that ε(p) = ε(r(g_{0})q) = (ε ⊗ g_{0})Δq = g_{0}(q) = 0, and for every g ∈ G(P) we have g(p) = (ε ⊗ g)Δp = ε(r(g)p) = 0. This implies G(𝔞) ⊆ G′(𝔞), and thus, finally, iii).

1Lecture delivered at the summer session of Seminar Sophus Lie in Darmstadt on May 14,

1.2. Consequences.

A. A Zariski-closed subsemigroup of an affine algebraic group over an infinite base field is an affine algebraic group.

B. The group of units of an affine algebraic semigroup is an affine algebraic group [6].

In the following we assume the base field to be algebraically closed. C. Let H denote an affine Hopf algebra (i.e., H is commutative and finitely generated). Every linear subspace 𝔞, which is an ideal and a coideal of H, is annihilated by the augmentation or counit ε of H, that is, ε(𝔞) = 0, and is invariant under the antipode or symmetry σ of H, that is σ(𝔞) ⊆ 𝔞. In short: Every biideal of an affine Hopf algebra is a Hopf ideal [5].

D. Every quotient bialgebra of an affine Hopf algebra is a quotient Hopf algebra.

E. If S is an affine algebraic semigroup, then for every s ∈ S there exists a power s^{k} which belongs to a subgroup of S [6]. There is an “elementary” proof of this statement, which is sketched as follows: Assume S ⊆ Mat(n; 𝕂) to be an affine algebraic semigroup. Denote by G′_{m} the group of n × n matrices of the form

( \begin{matrix}A & O \\ O & O \end{matrix} ); A ∈ GL(m; 𝕂), 1 ≤ m ≤ n,

G′_{m} ≅ GL(m; 𝕂). If s ∈ S and rank s = m, then there exists a matrix t ∈ GL(n; 𝕂) such that s′ = tst^{−1} has Jordan normal form. There is a power of s′ such that s^{′k} ∈ G′_{m}. The group G_{m} = t^{−1}G′_{m}t is isomorphic to GL(m; 𝕂) and H = S ∩ G_{m} is not void since s^{k} ∈ H. Also, H is a group by the theorem.

F. If S is an affine algebraic semigroup, then for every s ∈ S there exists an idempotent e_{0} such that for suitable powers of s we have e_{0}s^{k} = s^{k} = s^{k}e_{0} [6].

G. If there is only one idempotent in an affine algebraic monoid S, then S is a group [6].

2. Semialgebraic Semigroups

2.1. Foundations

From now on we take the base field 𝕂 to be the field of real numbers ℝ. We denote the Hopf algebra of polynomial functions on GL(n) = GL(n; ℝ) by

P(n) = ℝ[X_{11}, …, X_{nn}, x]/(x det X − 1).

Let G denote a real affine algebraic group

G = ann_{GL(n)} 𝔞, and 𝔞 = ann_{P(n)}G,

where 𝔞 denotes an ideal and a coideal of P(n), contained in the kernel of the augmentation ε, and invariant under the antipode σ of P(n)

Δ𝔞 ⊆ 𝔞 ⊗ P(n) + P(n) ⊗ 𝔞, ε(𝔞) = 0, σ(𝔞) ⊆ 𝔞.

Accordingly,

P = P(G) = P(n)/𝔞

denotes the Hopf algebra of polynomial functions on G.

Let S ⊆ G denote a subsemigroup of G. We call S a semialgebraic semigroup if

S = {s ∈ G; r_{j}(s) ≥ 0, r_{j} ∈ P, j = 1, …, l}.

Then S is a “closed” affine semialgebraic set. For simplicity we restrict the considerations to “closed” semialgebraic semigroups. If 𝔟 = ann_{P} S denotes the annihilator of S in P, then 𝔟 is an ideal and a coideal of P and by 1.2C it is contained in the kernel of ε and is invariant under σ, hence P^{∗} = P/𝔟 is a real Hopf algebra defining a real affine algebraic subgroup G^{∗} of G, which contains S as a subsemigroup. G^{∗} is the “smallest” real affine algebraic group which contains S as a subsemigroup.

In the sequel we may assume G = G^{∗}. Let S ⊆ G denote a closed affine semialgebraic set

S = {s ∈ G; r_{j}(s) ≥ 0, r_{j} ∈ P, j = 1, …, l}.

The cone

P_{+}(S) = {p ∈ P : (∀s ∈ S) p(s) ≥ 0}

defines a semiorder on P:

p ≤_{S} q iff q − p ∈ P_{+}(S).

We observe that p ≤_{S} q and q ≤_{S} p implies q − p ∈ 𝔟 = ann_{P}S. Let us identify

the tensor product P ⊗ P with the algebra of polynomial functions on G × G. The semialgebraic set S is a semialgebraic semigroup iff

i)  ΔP_{+}(S) ⊆ (P ⊗ P)_{+}(S × S),

it is a semialgebraic monoid iff, moreover, it is true that

ii)  ε(p) ≥ 0 for every p ∈ P_{+}(S),

and it is a semialgebraic group iff, in addition, the following condition is satisfied:

iii)  σ(P_{+}(S)) ⊆ P_{+}(S).

If S is a semialgebraic semigroup, then, by our assumption G = G^{∗}, the relations p ≤_{S} q and q ≤_{S} p imply p = q for every p, q ∈ P. Let G(P) denote the structure group of the Hopf algebra P. Then G(P) ≅ G canonically, and we have the isomorphism

S ≅ {ς ∈ G(P) : ς(P_{+}(S)) ⊆ ℝ_{+}}
= {ς ∈ G(P) : (∀p, q ∈ P) p ≤_{S} q implies ς(p) ≤ ς(q)}.

Assume G to be an irreducible real affine algebraic group, and denote by R = R(G) the field of fractions of the polynomial functions P = P(G). I.e., R is the field of rational functions on G. Set

S = {s ∈ G : r_{j}(s) ≥ 0, r_{j} ∈ P, (j = 1, …, l)}.

Then S is a semialgebraic subset of G. We denote by ∑ R^{2} the cone consisting of the sums of squares of elements from R, and by ∑ R^{2}(r_{j}; (j = 1, …, l)) the cone in R generated by r_{1}, …, r_{l}. The elements of the cone ∑ R^{2}(r_{j}; (j = 1, …, l)) are of the form f = f_{0} + ∑_{i=1}^{k} f_{i}t_{i}, where f_{i} ∈ ∑ R^{2}, i = 1, …, k, and where the t_{i}, (i = 1, …, k) are products of the r_{1}, …, r_{l}. The Positivstellensatz ([1], pp. 84, 93, 95) implies the following equation:

P_{+}(S) = ∑ R^{2}(r_{j}; (j = 1, …, l)) ∩ P.

Keeping in mind that the direct product G × G is irreducible, too, and using once more the Positivstellensatz we get the following criterion:

Let G denote an irreducible real affine algebraic group. A semialgebraic subset

S = {s ∈ G : r_{j}(s) ≥ 0, r_{j} ∈ P, j = 1, …, l}

of G is a semialgebraic semigroup iff

i)  Δr_{j} = f_{0j} + ∑_{i=1}^{k} f_{ij}(t_{ij} ⊗ t′_{ij}), j = 1, …, l,

where the f_{ij}, i = 0, 1, …, k, j = 1, …, l are sums of squares of elements from the field of fractions of P ⊗ P, and where the t_{ij}, t′_{ij}, i = 1, …, k, j = 1, …, l are products of the r_{1}, …, r_{l}.

Let us consider the case G = GL(n). Write R(n) for the field of fractions of P(n), then R(n) is isomorphic to R(X) = R(X_{11}, …, X_{nn}). Define a semialgebraic set by the equation

S = {s ∈ GL(n); r_{j}(s) ≥ 0, r_{j} ∈ P(n), j = 1, …, l},

then we may choose r_{j} ∈ R[X] = R[X_{11}, …, X_{nn}], j = 1, …, l. Now assume G = GL(n) × GL(n). Then P(G) ≅ P(n) ⊗ P(n). Denote the field of fractions by R(n, n). Now R(n, n) is isomorphic to R(X, Y) = R(X_{11}, …, X_{nn}, Y_{11}, …, Y_{nn}). Using the Positivstellensatz the criterion looks as follows:

The semialgebraic set

S = {s ∈ GL(n) : r_{j}(s) ≥ 0, r_{j} ∈ R[X], j = 1, …, l}

is a semialgebraic semigroup iff

i)  Δr_{j} = f_{0j} + ∑_{i=1}^{k} f_{ij}(t_{ij} ⊗ t′_{ij}) j = 1, …, l,

where the f_{ij}, i = 0, 1, …, k, j = 1, …, l are sums of squares of elements from R(n, n), f_{ij} ∈ ∑ R^{2}(n, n), and where the t_{ij}, t′_{ij}, i = 1, …, k, j = 1, …, l are products of the r_{1}, …, r_{l} ∈ R[X].

We may rewrite the necessary and sufficient condition i) using the isomorphisms indicated above and having in mind that the coproduct Δ is defined by matrix multiplication as follows:

i′)  r_{j}(XY) = f_{0j}(X, Y) + ∑_{i=1}^{k} f_{ij}(X, Y)t_{ij}(X)t′_{ij}(Y) j = 1, …, l.

Here r_{j}(XY) is a short notation for r_{j} (∑_{i=1}^{n} X_{1i}Y_{i1}, …, ∑_{i=1}^{n} X_{ni}Y_{in}); moreover, the functions f_{ij}(X, Y) = f_{ij}(X_{11}, …, X_{nn}, Y_{11}, …, Y_{nn}) belong to ∑ R(X, Y)^{2}, i.e., are sums of squares of rational functions in 2n^{2} variables, and the functions t_{ij}(X) = t_{ij}(X_{11}, …, X_{nn}) are products of the r_{1}(X), …, r_{l}(X), while the functions t′_{ij}(Y) = t′_{ij}(Y_{11}, …, Y_{nn}) are products of the r_{1}(Y), …, r_{l}(Y).

Let S denote a semialgebraic monoid. Assume S ⊆ G with a real affine algebraic group G, and denote by L(G) the Lie algebra of G. We define the Lie wedge of S by the following equation

W (S) = {u ∈ L(G); (∀p ∈ P_{+}(S), t ∈ ℝ_{+}) p(exp tu) ≥ 0}
= {u ∈ L(G); (∀t ∈ ℝ_{+}) r_{j}(exp tu) ≥ 0, j = 1, …, l}.

Evidently only those elements p ∈ P_{+}(S) are important, which satisfy the equation p(e) = ε(p) = 0. Regarding the r_{1}, …, r_{l} ∈ P defining the set S, we may assume r_{1}(e) = ⋯ = r_{k}(e) = 0, and r_{k+1}(e) > 0, …, r_{l}(e) > 0. Then

W (S) = {u ∈ L(G); (∀p ∈ P_{+}(S) with p(e) = 0, t ∈ ℝ_{+}) p(exp tu) ≥ 0}
= {u ∈ L(G); (∀t ∈ ℝ_{+}) r_{i}(exp tu) ≥ 0, i = 1, …, k}.

2.2 Examples

1) Let G = D(n) denote the group of regular real n × n diagonal matrices,

S = D_{+}^{m}(n) = {s ∈ D(n); s_{ii} ≥ 1, i = 1, …, m, s_{jj} ≥ 0, j = m + 1, …, n}.

The set D_{+}^{m}(n) is a semialgebraic monoid with the following Lie wedge:

W (D_{+}^{m}(n)) = {u ∈ R^{n^{2}}; u_{ij} = 0, i ≠ j, u_{ii} ≥ 0, i = 1, …, m}.

The group of units consists of the regular diagonal matrices having 1 at the first m places in the diagonal; it is isomorphic to D(n − m). Note that D^{0}(1) is the multiplicative group of positive real numbers, and that D^{1}(1) is the multiplicative monoid of real numbers ≥ 1.

2) Assume G = GL(n), and put

S = SL_{+}(n) = {s ∈ GL(n) : det s ≥ 1}.

Then SL_{+}(n) is a semialgebraic monoid with Lie wedge

W (SL_{+}(n)) = {u ∈ R^{n^{2}}; Tr u ≥ 0}.

The group of units of SL_{+}(n) is SL(n).

3) Assume G = GL(n), put

S = GL_{+}^{m}(n) = {s ∈ GL(n); s_{ii} ≥ 1, i = 1, …, m,
s_{ij} ≥ 0, i ≠ j or i = j = m + 1, …, n}.

The set GL_{+}^{m}(n) is a semialgebraic monoid with Lie wedge

W (GL_{+}^{m}(n)) = {u ∈ R^{n^{2}}; u_{ii} ≥ 0, i = 1, …, m, u_{ij} ≥ 0, i ≠ j}.

Now GL_{+}^{0}(n) is the monoid of nonnegative regular n × n matrices.

4) Define SL_{+}^{m}(n) = GL_{+}^{m}(n) ∩ SL_{+}(n). Then SL_{+}^{m}(n) is a semialgebraic monoid and

W (SL_{+}^{m}(n)) = W (GL_{+}^{m}(n)) ∩ W (SL_{+}(n)).

In the case m = n we have

W (SL_{+}^{n}(n)) = W (GL_{+}^{n}(n)).

5) By s ( \begin{matrix}i_{1} ⋯ i_{k} \\ j_{1} ⋯ j_{k} \end{matrix} ) we denote the k -minor of the matrix s given by the rows with

indices i_{1} ⋯ i_{k} and the columns with the indices j_{1} ⋯ j_{k}. Assume G = GL(n), and put

S = GL_{++}^{m}(n) = \{s ∈ GL(n); s ( \begin{matrix}i_{1} ⋯ i_{k} \\ i_{1} ⋯ i_{k} \end{matrix} ) \{ \begin{matrix}≥ 1 & if 1 ≤ i_{1} ≤ ⋯ ≤ i_{k} ≤ m, \\ ≥ 0 & otherwise \end{matrix}\} .

The set GL_{++}^{m}(n) is a semialgebraic monoid. The Lie wedge of GL_{++}^{m}(n) equals the Lie wedge of GL_{+}^{m}(n). Here GL_{++}^{0}(n) is the monoid of completely nonnegative regular n × n matrices. If n ≥ 3 then

GL_{++}^{n}(n) ⊂ SL_{+}^{n}(n) ⊂ GL_{+}^{n}(n),

and

W (GL_{++}^{n}(n)) = W (SL_{+}^{n}(n)) = W (GL_{+}^{n}(n)).

6) Let G denote the group of “similarities” in euclidean n-space:

G = {g ∈ GL(n); g = αh, α ∈ R, α ≠ 0, and h ∈ SO(n)}.

We define

S = {s ∈ G; s = αh, α ≥ 1, and h ∈ SO(n)}.

The set S is a semialgebraic monoid with the Lie wedge

W (S) = {αu ∈ R^{n^{2}}; α ≥ 0 and u ∈ so(n)}.

S is the monoid of “expanding” similarities, isomorphic to the direct product of the monoid D^{1}(1) and the orthogonal group SO(n) : S ≅ D^{1}(1)×SO(n); its Lie wedge is isomorphic to ℝ_{+} ⊕ so(n); its unit group is SO(n).

7) The last example is a little bit beyond the context of this lecture. Consider the set

S = Osc(n) = {s ∈ GL_{++}^{0}(n); s_{ij} > 0, if |i − j| ≤ 1}.

The set Osc(n) is the semialgebraic semigroup of oscillation matrices ([3], p.447); it is not a monoid, and it is not “closed”.

References

[1] J. Bochnak, M. Coste, M.-F. Roy, “Géometrie algébrique réelle”, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge, Band 12, Springer Verlag 1987.

[2] C. Chevalley, “Théorie des Groupes de Lie, Tome II, Groupes Algébriques,” Hermann & Cie. Paris 1951.

[3] F. R. Gantmacher, ,,Matrizentheorie“, Deutscher Verlag d. Wiss. Berlin 1986.

[4] K. H. Hofmann, A short course on the Lie theory of semigroups I : The fundamental theorems. Seminar Sophus Lie 1 (1991), 33–40.

[5] K. Newman, A Correspondence between Bi-ideals and Sub-Hopf Algebras in Cocommutative Hopf Algebras, Journal of Algebra 36 (1975), 1–15.

[6]  M. S. Putcha, “Linear algebraic monoids,” Cambridge Univ. Press 1988.

Fachrichtungen Mathematik/Informatik Ernst Moritz Arndt Universität Greifswald Friedrich Ludwig Jahn-Straße 15a D-17487 Greifswald, Germany boseck@math-inf.uni-greifswald.dbp.de

Received August 17, 1993