Finite-dimensional Semisimple Algebra Research Articles

Potentials for some tensor algebras

This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras E over a field F, such that E⊗FEop is semisimple. We assume that E contains a certain type of F-basis which is a generalization of a multiplicative basis. We study potentials belonging to the algebra of formal power series, with coefficients in the tensor algebra over E, of any finite-dimensional E-E-bimodule on which F acts centrally. In this case, we introduce a cyclic derivative and to each potential we associate a Jacobian ideal. Finally, we develop a mutation theory of potentials, which in the case that the bimodule is Z-free, it behaves as the quiver case; but allows us to obtain realizations of a certain class of skew-symmetrizable integer matrices.

The QRD and SVD of matrices over a real algebra

Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD to a wide class of real algebras, including all finite-dimensional semi-simple algebras, (twisted) group algebras and Clifford algebras. Two approaches are described for computing the QRD/SVD: one Jacobi method with a generalised Givens rotation, and one based on the Artin–Wedderburn theorem.

A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph - which we name as a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant {\kappa}(A) for A and count the number of isomorphism classes of Leavitt path algebras with {\kappa}(A)=n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.

The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring $k\langle x,y\rangle /(y^2)$

It is shown that the noncommutative algebraic curve with homogeneous coordinate ring \mathbb{C}\langle x,y\rangle/(y^2) is a noncommutative algebraic-geometric analogue of the space of Penrose tilings of the plane. Individual tilings determine “points” on the noncommutative curve and the tilings coincide under isometry if and only if the skyscraper sheaves of the corresponding points are isomorphic. The category of quasi-coherent sheaves on the curve is equivalent to the category of modules over a von Neumann regular ring that is a direct limit of finite dimensional semisimple algebras. The norm closure of this von Neumann regular ring is the AF-algebra that Connes associates to the space of Penrose tilings. There is an algebraic analogue of the fact that every isometry-invariant subset of tilings is dense in the set of all Penrose tilings.

On the codimension growth of almost nilpotent Lie algebras

We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra $L$ is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of $L$ exists and is a positive integer.

Depth One Extensions of Semisimple Algebras and Hopf Subalgebras

Depth one extensions of finite dimensional semisimple algebras are completely characterized in terms of their algebra centers. For extensions of semisimple Hopf algebras this characterization translates into a trivial monoidal action of the dual fusion category Rep(A*) on Rep(B).

Category equivalences involving graded modules over path algebras of quivers

Let Q be a finite quiver with vertex set I and arrow set Q1, k a field, and kQ its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr(kQ)≔Gr(kQ)/Fdim(kQ) of graded kQ-modules modulo those that are the sum of their finite dimensional submodules, namely QGr(kQ)≡ModS(Q)≡GrL(Q∘)≡ModL(Q∘)0≡QGr(kQ(n)). Here S(Q)=lim⟶EndkI(kQ1⊗n) is a direct limit of finite dimensional semisimple algebras; Q∘ is the quiver without sources or sinks that is obtained by repeatedly removing all sinks and sources from Q; L(Q∘) is the Leavitt path algebra of Q∘; L(Q∘)0 is its degree zero component; and Q(n) is the quiver whose incidence matrix is the nth power of that for Q. It is also shown that all short exact sequences in qgr(kQ), the full subcategory of finitely presented objects in QGr(kQ), split. Consequently qgr(kQ) can be given the structure of a triangulated category with suspension functor the Serre degree twist (−1); this triangulated category is equivalent to the “singularity category” Db(Λ)/Dperf(Λ) where Λ is the radical square zero algebra kQ/kQ≥2, and Db(Λ) is the bounded derived category of finite dimensional left Λ-modules.

The stable rank of arithmetic orders in division algebras – an elementary approach

A well-known theorem of Bass implies that 2 defines a stable range for an arithmetic order in a finite-dimensional semisimple algebra over an algebraic number field. The purpose of this note is to provide an independent and elementary proof of this fact for arithmetic orders contained in a finite-dimensional division algebra over an algebraic number field.

Lie algebras and algebras of associative type

In the paper, some properties of algebras of associative type are studied, and these properties are then used to describe the structure of finite-dimensional semisimple modular Lie algebras. It is proved that the homogeneous radical of any finite-dimensional algebra of associative type coincides with the kernel of some form induced by the trace function with values in a polynomial ring. This fact is used to show that every finite-dimensional semisimple algebra of associative type A = ⊕αeG A α graded by some group G, over a field of characteristic zero, has a nonzero component A 1 (where 1 stands for the identity element of G), and A 1 is a semisimple associative algebra. Let B = ⊕αeG B α be a finite-dimensional semisimple Lie algebra over a prime field F p , and let B be graded by a commutative group G. If B = F p ⊗ ℤ A L , where A L is the commutator algebra of a ℤ-algebra A = ⊕αeG A α ; if ℚ ◯ ℤ A is an algebra of associative type, then the 1-component of the algebra K ◯ ℤ B, where K stands for the algebraic closure of the field F p , is the sum of some algebras of the form gl(n i ,K).

Representations of finite association schemes

This article is a survey on representation theory of association schemes including recent developments and some applications. A lot of known formulas on complex characters are also obtained as corollaries to results on finite dimensional semisimple algebras. Clifford theory of association schemes and related results are explained. Also this article contains basic parts of modular representation theory. Modular representation theory is new and remarkable method in this area. Some open questions and related results are given.

Double Poisson Structures on Finite Dimensional Semi-Simple Algebras

We give a description of the bimodule of double derivations of a finite dimensional semi-simple algebra S and its double Schouten bracket in terms of a quiver. This description is used to determine which degree two monomials induce double Poisson brackets on S. In case S = ℂ⊕n , a criterion for any degree two element to give a double Poisson bracket is deduced. For S = ℂ⊕n and S′ = ℂ⊕m the induced Poisson bracket on the variety of isomorphism classes of semi-simple representations iss n (S * T) of the free product S * T is given.

Beyond the Alder–Strassen bound

We prove a lower bound of 5 2 n 2 - 3 n for the multiplicative complexity of n × n -matrix multiplication over arbitrary fields. More general, we show that for any finite dimensional semisimple algebra A with unity, the multiplicative complexity C ( A ) of the multiplication in A is bounded from below by 5 2 dim A - 3 ( n 1 + ⋯ + n t ) if the decomposition of A ≅ A 1 × ⋯ × A t into simple algebras A τ ≅ D τ n τ × n τ contains only noncommutative factors, that is, the division algebra D τ is noncommutative or n τ ⩾ 2 . We also deal with the complexity of multiplication in algebras with nonzero radical. We present an example that shows that our methods in the semisimple case cannot be applied directly to this problem. We exhibit lower bound techniques for C ( A ) that yield bounds still significantly above the Alder–Strassen bound. The main application is the lower bound C ( T n ( k ) ) ⩾ ( 2 1 8 - o ( 1 ) ) dim T n ( k ) for the multiplicative complexity of multiplication in the algebra T n ( k ) of upper triangular n × n -matrices.

Contractible Fréchet algebras

A unital Frechet algebra A is called contractible if there exists an element d such that π A (d) = 1 and ad = da for all a ∈ A where π A : A⊗A → A is the canonical Frechet A-bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Frechet algebra A to be a direct sum of a finite-dimensional semisimple algebra M and a contractible Frechet algebra N without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative Imc Frechet Q-algebra is contractible if, and only if, it is algebraically and topologically isomorphic to C n for some n ∈ N. On the other hand, we show that a Frechet algebra, that is, a locally C*-algebra, is contractible if, and only if, it is topologically isomorphic to the topological Cartesian product of a certain countable family of full matrix algebras.

Semiprime Graded Algebras of Dimension Two

Semiprime, noetherian, connected graded k-algebras R of quadratic growth are described in terms of geometric data. A typical example of such a ring is obtained as follows: Let Y be a projective variety of dimension at most one over the base field k and let E be an OY-order in a finite dimensional semisimple algebra A over K=k(Y). Then, for any automorphism τ of A that restricts to an automorphism σ of Y and any ample, invertible E-bimodule B, Van den Bergh constructs a noetherian, “twisted homogeneous coordinate ring” B=⊕H0(Y,B⊗···⊗Bτn−1). We show that R is noetherian if and only if some Veronese ring R(m) of R has the form k+I, where I is a left ideal of such a ring B and where I=B at each point p∈Y at which σ has finite order. This allows one to give detailed information about the structure of R and its modules.

Vertex Operator Algebras and Associative Algebras

LetVbe a vertex operator algebra. We construct a sequence of associative algebrasAn(V) (n=0,1,2,…) such thatAn(V) is a quotient ofAn+1(V) and a pair of functors between the category ofAn(V)-modules which are notAn−1(V)-modules and the category of admissibleV-modules. These functors exhibit a bijection between the simple modules in each category. We also show thatVis rational if and only if allAn(V) are finite-dimensional semisimple algebras.

Twisted representations of vertex operator algebras and associative algebras

Let V be a vertex operator algebra and g an automorphism of order T. We construct a sequence of associative algebras A_{g,n}(V) with n\in\frac{1}{T}\Z nonnegative such that A_{g,n}(V) is a quotient of A_{g,n+1/T}(V) and a pair of functors between the category of A_{g,n}(V)-modules which are not A_{g,n-1/T}(V)-modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is g-rational if and only if all A_{g,n}(V) are finite-dimensional semisimple algebras.

Minimal number of idempotent generators of matrix algebras over arbitrary field

It is proved that the smallest number v=(n,F) such that the matrix algebra Mn(F) (n> 2) over an arbitrary field F can be generated (as an algebra) by v idempotents is for the remaining cases. The minimal number of idempotent generators of a split finite-dimensional semi-simple algebra is also obtained.

Towers of semi-simple algebras

This is an extension of the work of Goodman-de la Harpe-Jones on pairs of multi-matrix algebras and the corresponding index. In particular, it is shown that the basic invariant of a pair of finite-dimensional semi-simple algebras is a bimodule together with an integral n-tuple, and that the entire theory is equivalent to the theory of finite-dimensional hereditary algebras whose square of the radical equals zero.

A study of W-algebras using Jacobi identities

A theoretical framework in which to study the Jacobi identities on the commulators of W-algebra fields is established. W-algebras containing various specified combinations of fields of spin up to 8 are classified and constructed. In the cases studied, we show that there are closed W-algebraic structures, valid for all values of the central charge c, only for those field contents corresponding to the exponents of finite-dimensional semi-simple algebras.

The scheme of finite-dimensional representations of an algebra

For a finitely generated /^-algebra A and a finite dimension- al ^-vector space M the representations of A on M form an affine ^-scheme Mod^(M). Of particular interest for this scheme are the connected components, the irreducible components, and the open and closed orbits under the natural action of the general linear group AutA(M), since the orbits are the equiva- lence classes of representatio ns. The connected components are known for a finite dimensional algebra A. In this paper we characterize the connected components when A is com- mutative or an enveloping algebra of a Lie algebra in chara- cteristic zero. For the algebra k(x> y)/(x, y)2 we describe the open orbits and the irreducible components. Finally, we ex- amine the connection with the theory of deformations of al- gebra representations. Introduction* For almost any ^-algebra A it is an impossible task to classify all finite dimensional A-modules. There are some standard exceptions such as the finite dimensional semisimple algebras, A = k(x)/(xm), A = k(x, y)/(x, yf, and A = k(x). Even for the polynomial algebra in two indeterminates, k(x, y), Gelfand and Ponomarev have shown that the classification problem is as difficult as classifying the modules over the free algebra k{x19 •••, xm}. (See (7) for details.) If we restrict our attention to modules of a given dimension, say n, then to classify the ^-dimensional A-modules is to classify the orbits of Aut}χkn) = GL(n) acting on the space of ^-dimensional A-module structures. If we take A to be a finitely generated k- algebra then this space of module structures is an affine fc-scheme, which we call Mod^(M) where M = kn. Although we cannot hope to determine the orbits in most cases, we may be able to describe coarser features of ModA(M) such as the irreducible components and the connected components. It is surprising that even for the con- nected components there is not a complete answer for an arbitrary finitely generated ^-algebra, while the irreducible components are not at all understood. Only for one interesting algebra, namely A = k(x, y)/(x, y)2, have the irreducible components been determined, and for this it is essential to know the classification of the finite dimensional A-modules. (In § 5 we recall the description of the indecomposable modules from (9), determine all open orbits, and from them describe the irreducible components as found by Flanigan