How to compute the Wedderburn decomposition of a finite-dimensional associative algebra

In this paper we propose a general recipe for calculating the automorphism groups of semigroups consisting of partial endomorphisms of relational structures over a finite set with a single m-ary relation for any m E N. We use this recipe to determine the automorphism groups of the following semigroups: the full transformation semigroup, the partial transformation semigroup, and the symmetric inverse semigroup, the wreath product of two full transformation semigroups, the partial endomorphisms of any partially ordered set, the full spectrum of semigroups of partial mappings preserving or reversing a linear or circular order.

This chapter studies finite-dimensional associative division algebras, as well as other finite-dimensional associative algebras and closely related rings. The chapter is in two parts that overlap slightly in Section 6. The first part gives the structure theory of the rings in question, and the second part aims at understanding limitations imposed by the structure of a division ring. Section 1 briefly summarizes the structure theory for finite-dimensional (nonassociative) Lie algebras that was the primary historical motivation for structure theory in the associative case. All the algebras in this chapter except those explicitly called Lie algebras are understood to be associative. Section 2 introduces left semisimple rings, defined as rings $R$ with identity such that the left $R$ module $R$ is semisimple. Wedderburn's Theorem says that such a ring is the finite product of full matrix rings over division rings. The number of factors, the size of each matrix ring, and the isomorphism class of each division ring are uniquely determined. It follows that left semisimple and right semisimple are the same. If the ring is a finite-dimensional algebra over a field $F$, then the various division rings are finite-dimensional division algebras over $F$. The factors of semisimple rings are simple, i.e., are nonzero and have no nontrivial two-sided ideals, but an example is given to show that a simple ring need not be semisimple. Every finite-dimensional simple algebra is semisimple. Section 3 introduces chain conditions into the discussion as a useful generalization of finite dimensionality. A ring $R$ with identity is left Artinian if the left ideals of the ring satisfy the descending chain condition. Artin's Theorem for simple rings is that left Artinian is equivalent to semisimplicity, hence to the condition that the given ring be a full matrix ring over a division ring. Sections 4–6 concern what happens when the assumption of semisimplicity is dropped but some finiteness condition is maintained. Section 4 introduces the Wedderburn–Artin radical $\mathrm{rad} R$ of a left Artinian ring $R$ as the sum of all nilpotent left ideals. The radical is a two-sided nilpotent ideal. It is 0 if and only if the ring is semisimple. More generally $R/\mathrm{rad} R$ is always semisimple if $R$ is left Artinian. Sections 5–6 state and prove Wedderburn's Main Theorem–-that a finite-dimensional algebra $R$ with identity over a field $F$ of characteristic 0 has a semisimple subalgebra $S$ such that $R$ is isomorphic as a vector space to $S\oplus\mathrm{rad} R$. The semisimple algebra $S$ is isomorphic to $R/\mathrm{rad} R$. Section 5 gives the hard part of the proof, which handles the special case that $R/\mathrm{rad} R$ is isomorphic to a product of full matrix algebras over $F$. The remainder of the proof, which appears in Section 6, follows relatively quickly from the special case in Section 5 and an investigation of circumstances under which the tensor product over $F$ of two semisimple algebras is semisimple. Such a tensor product is not always semisimple, but it is semisimple in characteristic 0. The results about tensor products in Section 6, but with other hypotheses in place of the condition of characteristic 0, play a role in the remainder of the chapter, which is aimed at identifying certain division rings. Sections 7–8 provide general tools. Section 7 begins with further results about tensor products. Then the Skolem–Noether Theorem gives a relationship between any two homomorphisms of a simple subalgebra into a simple algebra whose center coincides with the underlying field of scalars. Section 8 proves the Double Centralizer Theorem, which says for this situation that the centralizer of the simple subalgebra in the whole algebra is simple and that the product of the dimensions of the subalgebra and the centralizer is the dimension of the whole algebra. Sections 9–10 apply the results of Sections 6–8 to obtain two celebrated theorems—Wedderburn's Theorem about finite division rings and Frobenius's Theorem classifying the finite-dimensional associative division algebras over the reals.

The first structure theory in abstract algebra was that of finite dimensional Lie algebras (Cartan-Killing), followed by the structure theory of associative algebras (Wedderburn-Artin). These theories determine, in a non-constructive way, the basic building blocks of the respective algebras (the radical and the simple components of the factor by the radical). In view of the extensive computations done in such algebras, it seems important to design efficient algorithms to find these building blocks.We find polynomial time solutions to a substantial part of these problems. We restrict our attention to algebras over finite fields and over algebraic number fields. We succeed in determining the radical (the “bad part” of the algebra) in polynomial time, using (in the case of prime characteristic) some new algebraic results developed in this paper. For associative algebras we are able to determine the simple components as well. This latter result generalizes factorization of polynomials over the given field. Correspondingly, our algorithm over finite fields is Las Vegas.Some of the results generalize to fields given by oracles.Some fundamental problems remain open. An example: decide whether or not a given rational algebra is a noncommutative field.

Marques-Smith and Sullivan [‘Partial orders on transformation semigroups’, Monatsh. Math.140 (2003), 103–118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the ‘containment order’: namely, if α,β∈P(X) then α⊆β means xα=xβ for all x∈dom α, the domain of α. The other order was the so-called ‘natural order’ defined by Mitsch [‘A natural partial order for semigroups’, Proc. Amer. Math. Soc.97(3) (1986), 384–388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer–Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.

The diagonal right (respectively, left) act of a semigroup S is the set S × S on which S acts via (x, y) s = (xs, ys) (respectively, s (x, y) = (sx, sy)); the same set with both actions is the diagonal bi-act. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set A ⊆ S × S such that S × S = AS1 (respectively, S × S = S1A, S × S = SlASl).In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semi-groups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts.

Factorization of polynomials over finite fields and decomposition of primes in algebraic number fields

We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.

For an integer let be the symmetric inverse semigroup of partial injective transformations on an n-element set. For denote by and respectively, the first and second centralizer of λ in We determine the structure of and in terms of Green’s relations, including the partial order of -classes, and express as a direct product of cyclic groups with zero adjoined and a monogenic monoid. For each individual Green relation we determine such that in is inherited from and λ such that all Green relations in are inherited from We also provide a representation of the partial order of -classes in as a known lattice, and describe such that which gives a class of maximal commutative subsemigroups of

The commuting graph of a finite non-commutative semigroup $S$, denoted $\cg(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let $\mi(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup $\mi(X)$ has the symmetric group $\sym(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\sym(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of $\cg(\sym(X))$, and in 2011, Dol\u{z}an and Oblak claimed (but their proof has a GAP) that this upper bound is in fact the exact value. The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup $\mi(X)$. We calculate the clique number of $\cg(\mi(X))$, the diameters of the commuting graphs of the proper ideals of $\mi(X)$, and the diameter of $\cg(\mi(X))$ when $|X|$ is even or a power of an odd prime. We show that when $|X|$ is odd and divisible by at least two primes, then the diameter of $\cg(\mi(X))$ is either 4 or 5. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of $\mi(X)$ of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of $\mi(X)$. The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.

Let $$\mathcal {I}(X)$$ be the symmetric inverse semigroup of partial injective transformations on a set X (finite or infinite). For $$\alpha \in \mathcal {I}(X)$$ , let $$C(\alpha )=\{\beta \in \mathcal {I}(X):\alpha \beta =\beta \alpha \}$$ be the centralizer of $$\alpha $$ in $$\mathcal {I}(X)$$ . Consider $$\alpha \in \mathcal {I}(X)$$ with $${{\mathrm{dom}}}(\alpha )=X$$ . For each Green relation $$\mathcal {G}$$ , we determine $$\alpha $$ such that $$\mathcal {G}$$ in $$C(\alpha )$$ is the restriction of the corresponding relation in $$\mathcal {I}(X)$$ ; $$\alpha $$ such that all Green relations in $$C(\alpha )$$ are the restrictions of the corresponding relations in $$\mathcal {I}(X)$$ ; $$\alpha $$ for which $$\mathcal {D}=\mathcal {J}$$ in $$C(\alpha )$$ ; $$\alpha $$ for which the partial order of $$\mathcal {J}$$ -classes in $$C(\alpha )$$ is the restriction of the corresponding partial order in $$\mathcal {I}(X)$$ ; and finally $$\alpha $$ for which the $$\mathcal {J}$$ -classes in $$C(\alpha )$$ are totally ordered. The descriptions are in terms of the cycle-ray decomposition of $$\alpha $$ , which is a generalization of the cycle decomposition of a permutation.

The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

Space-time codes for a wide variety of channels have the property that the diversity of a pair of codeword matrices is measured by the vanishing or non-vanishing of polynomials in the entries of the matrices. We show that for every such channel: I) There is an appropriately-defined notion of approximation of space-time codes such that each code is arbitrarily well approximated by one whose alphabet lies in the field of algebraic numbers; II) Each space-time code whose alphabet lies in the field of algebraic numbers is an appropriately-defined lift from a corresponding space-time code defined over a finite field or a ''scaled'' lift from a Galois ring of arbitrary characteristic. This implies that all space-time codes can be designed over finite fields or over Galois rings of arbitrary characteristic and then lifted to complex matrices with entries in a number field.

The large rank of a finite semigroup [Formula: see text] is the least number [Formula: see text] such that every subset of [Formula: see text] with [Formula: see text] elements generates [Formula: see text]. This paper obtains the large ranks of [Formula: see text] and [Formula: see text], the semigroups of singular transformations, injective partial and partial transformations on a finite chain [Formula: see text], which preserve or reverse the order, respectively. As a consequence, we obtain the large ranks of [Formula: see text] and [Formula: see text], the semigroups of injective order-preserving partial transformations and order-preserving partial transformations on [Formula: see text], respectively.

Computing the structure of finite algebras

Zero divisors in quaternion algebras