Optimal filtrations on representations of finite-dimensional algebras TeX Export

@article{citeulike:462883, abstract = { Let \$A\$ be a finite-dimensional algebra over an algebraically closed field \$k\$. Then \$A\$ can be identified with \$kQ/I\$ where \$Q\$ is a finite quiver and \$I\$ is an admissible ideal of the path algebra \$kQ\$. Moreover, the affine algebraic variety \${\rm Rep}(A,\alpha)\$ formed by all finite-dimensional representations of \$A\$ with fixed dimension vector \$\alpha=(a\sb 1,\cdots,a\sb n)\$ can be identified with a \${\rm GL}(\alpha)\$-subvariety of the affine space \${\rm Rep}(Q,\alpha)\$ of all \$\alpha\$-dimensional representations of \$Q\$, and the orbits correspond to isomorphism classes of \$A\$-representations. <P> The author observes that \${\rm Rep}(A,\alpha)\$ is a subvariety of the nullcone \${\rm Null}(Q,\alpha)\$ of \${\rm Rep} (Q,\alpha)\$ under the action of \${\rm GL}(\alpha)\$. So, he considers the Hesselink stratification of \${\rm Null}(Q,\alpha)\$ \ref[W. H. Hesselink, Invent. Math. <strong>55</strong> (1979), no. 2, 141--163; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=553706\&loc=fromrevtext">MR0553706 (81b:14025)</A>] and gives an explicit representation-theoretic description of the non-empty strata using the notion of a semistable representation introduced by A. D. King \ref[Quart. J. Math. Oxford Ser. (2) <strong>45</strong> (1994), no. 180, 515--530; <A HREF="/msnmain?fn=105\&fmt=doc\&r=1\&pg1=CNO\&s1=1315461\&loc=fromrevtext">MR1315461 (96a:16009)</A>]. He then applies these investigations to parametrize the isomorphism classes of uniserial \$A\$-representations with a fixed sequence of simple composition factors, recovering some recent results of K. Bongartz and B. Huisgen-Zimmermann ["Varieties of uniserial representations. IV. Kinship to geometric quotients", Preprint, Univ. California, Santa Barbara, CA, 1997; per bibl.]. He also points out that the same classification strategy applies to a more general situation, namely, to \$A\$-representations \$V\$ with certain optimal filtrations, that is, filtrations corresponding to certain optimal one-parameter subgroups for \$V\$ in \${\rm GL}(\alpha)\$.}, author = {Le Bruyn, Lieven}, citeulike-article-id = {462883}, citeulike-linkout-0 = {http://www.ams.org/mathscinet-getitem?mr=1707199}, journal = {Trans. Amer. Math. Soc.}, keywords = {algebras, filtrations, representations, stratification}, mrnumber = {MR1707199}, number = {1}, pages = {411--426}, posted-at = {2006-01-12 11:04:37}, priority = {0}, title = {Optimal filtrations on representations of finite-dimensional algebras}, url = {http://www.ams.org/mathscinet-getitem?mr=1707199}, volume = {353}, year = {2001} }

TY - JOUR ID - citeulike:462883 L3 - citeulike-article-id:462883 N2 - Let $A$ be a finite-dimensional algebra over an algebraically closed field $k$. Then $A$ can be identified with $kQ/I$ where $Q$ is a finite quiver and $I$ is an admissible ideal of the path algebra $kQ$. Moreover, the affine algebraic variety ${\rm Rep}(A,\alpha)$ formed by all finite-dimensional representations of $A$ with fixed dimension vector $\alpha=(a\sb 1,\cdots,a\sb n)$ can be identified with a ${\rm GL}(\alpha)$-subvariety of the affine space ${\rm Rep}(Q,\alpha)$ of all $\alpha$-dimensional representations of $Q$, and the orbits correspond to isomorphism classes of $A$-representations. <P> The author observes that ${\rm Rep}(A,\alpha)$ is a subvariety of the nullcone ${\rm Null}(Q,\alpha)$ of ${\rm Rep} (Q,\alpha)$ under the action of ${\rm GL}(\alpha)$. So, he considers the Hesselink stratification of ${\rm Null}(Q,\alpha)$ \ref[W. H. Hesselink, Invent. Math. <strong>55</strong> (1979), no. 2, 141--163; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=553706&loc=fromrevtext">MR0553706 (81b:14025)</A>] and gives an explicit representation-theoretic description of the non-empty strata using the notion of a semistable representation introduced by A. D. King \ref[Quart. J. Math. Oxford Ser. (2) <strong>45</strong> (1994), no. 180, 515--530; <A HREF="/msnmain?fn=105&fmt=doc&r=1&pg1=CNO&s1=1315461&loc=fromrevtext">MR1315461 (96a:16009)</A>]. He then applies these investigations to parametrize the isomorphism classes of uniserial $A$-representations with a fixed sequence of simple composition factors, recovering some recent results of K. Bongartz and B. Huisgen-Zimmermann ["Varieties of uniserial representations. IV. Kinship to geometric quotients", Preprint, Univ. California, Santa Barbara, CA, 1997; per bibl.]. He also points out that the same classification strategy applies to a more general situation, namely, to $A$-representations $V$ with certain optimal filtrations, that is, filtrations corresponding to certain optimal one-parameter subgroups for $V$ in ${\rm GL}(\alpha)$. IS - 1 JF - Trans. Amer. Math. Soc. EP - 426 TI - Optimal filtrations on representations of finite-dimensional algebras VL - 353 SP - 411 KW - algebras KW - filtrations KW - representations KW - stratification AU - Le Bruyn, Lieven PY - 2001/// UR - http://www.ams.org/mathscinet-getitem?mr=1707199 ER -